Title: Active Preload Control of a Redundantly Actuated Stewart Platform for Backlash Prevention

Transcription

1 Elsevier Editorial System(tm) for Robotics and Computer Integrated Manufacturing Manuscript Draft Manuscript Number: Title: Active Preload Control of a Redundantly Actuated Stewart Platform for Backlash Prevention Article Type: Research Paper Keywords: Backlash prevention; Stewart platform; redundant actuation; active preload control; online optimization Corresponding Author: Mr. Boyin Ding, Ph.D student Corresponding Author's Institution: University of Adelaide First Author: Boyin Ding, Ph.D student Order of Authors: Boyin Ding, Ph.D student; Benjamin S Cazzolato, Ph.D; Steven Grainger, Ph.D; Richard M Stanley, Engineer; John J Costi, Ph.D Abstract: There is an increasing trend to use Stewart platforms to implement ultra-high precision tasks under large interactive loads (e.g. machining, material testing) mainly due to their high stiffness, and high load carrying capacity. However, the backlash or joint clearance in the system can significantly degrade the accuracy and bandwidth. This work studied the application of actuation redundancy in a general Stewart platform to regulate the preloads on its active joints for the purpose of backlash prevention. A novel active preload control method was proposed to achieve a real-time approach that is robust to large six degree of freedom interactive loads. The proposed preload method applies an inverse-dynamics based online optimization algorithm to calculate the desired force trajectory of the redundant actuator, and uses a force control scheme to achieve the required force. Simulation and experimental results demonstrate that this method is able to eliminate backlash inaccuracies during application of large interactive loads and therefore ensure the precision of the system.

2 Cover Letter MECHANICAL ENGINEERING FACULTY OF ENGINEERING, COMPUTER AND MATHEMATICAL SCIENCES BOYIN DING MECHANICAL ENGINEERING THE UNIVERSITY OF ADELAIDE SA 5005 AUSTRALIA TELEPHONE FACSIMILE boyin.ding@adelaide.edu.au CRICOS Provider Number 00123M 10th Oct 2013 Re: Research paper submission Dear Editor We would like you consider the attached paper entitled Active preload control of a redundantly actuated Stewart platform for backlash prevention for submission to the Journal of Robotics and Computer Integrated Manufacturing. We certify that this article is original, that it is not under consideration by another journal or been previously published. All named authors were involved in the conception of the idea, data collection, data analysis and drafting of the final manuscript. Yours sincerely, Boyin Ding School of Mechanical Engineering

3 *Highlights (for review) An online active preload control method with actuation redundancy was proposed to prevent backlash on a Stewart platform. The arrangement of the redundantly actuated manipulator demonstrated effective active preload distribution efficiency, particularly when placing the redundant leg into the robot inner space. The proposed preload control method significantly mitigated backlash limit cycles and consequently higher bandwidth control can be achieved on the robot with higher accuracy. The proposed method was robust to large six degrees of freedom interactive loads.

4 *Manuscript Click here to view linked References Manuscript for Robotics and Computer Integrated Manufacturing Active Preload Control of a Redundantly Actuated Stewart Platform for Backlash Prevention Boyin Ding a, Benjamin S. Cazzolato a, Steven Grainger a, Richard M. Stanley b, and John J. Costi b a School of Mechanical Engineering, University of Adelaide, Adelaide, SA 5005, Australia b Biomechanics & Implants Research Group, Medical Device Research Institute and School of Computer Science, Engineering and Mathematics, Flinders University, Bedford Park, SA 5042, Australia Correspondence author: Boyin Ding Permanent address: School of Mechanical Engineering, University of Adelaide, Adelaide, SA 5005, Australia address: boyin.ding@adelaide.edu.au Telephone number: +61 (08) Fax number: +61 (08)

5 Active preload control of a redundantly actuated Stewart platform for backlash prevention Abstract There is an increasing trend to use Stewart platforms to implement ultra-high precision tasks under large interactive loads (e.g. machining, material testing) mainly due to their high stiffness and high load carrying capacity. However, the backlash or joint clearance in the system can significantly degrade the accuracy and bandwidth. This work studied the application of actuation redundancy in a general Stewart platform to regulate the preloads on its active joints for the purpose of backlash prevention. A novel active preload control method was proposed to achieve a real-time approach that is robust to large six degree of freedom interactive loads. The proposed preload method applies an inverse-dynamics based online optimization algorithm to calculate the desired force trajectory of the redundant actuator, and uses a force control scheme to achieve the required force. Simulation and experimental results demonstrate that this method is able to eliminate backlash inaccuracies during application of large interactive loads and therefore ensure the precision of the system. Keywords: Backlash prevention, Stewart platform, redundant actuation, active preload control, online optimization 2

6 1. Introduction Parallel robots are well known for their advantages in providing higher rigidity and stiffness, being more compact in structure, and having greater payload capacity than their serial counterparts. As a result, they are often used in applications where precision of the order of micrometres is required from the robot during interaction (e.g. manufacturing and assembling). However, joint clearances or backlash can largely degrade the accuracy of parallel robots in these applications [1,2] as well as severely limiting bandwidth. Many linear and non-linear control methods have been proposed to mitigate backlash inaccuracies on a single actuated joint [3]. These methods often require a highly accurate backlash model which is difficult to approximate in practice. Flexure joints have been developed to remove backlash at the expense of limited range of motion [4,5]. Recent research found it was possible to achieve backlash prevention for parallel robots by controlling the preloads on their actuated joints. Preload control can be further divided into two categories: the active method and the passive method. The active method uses actuation redundancy while considering dynamic effects for the purpose of backlash prevention along a specified path [6,7]. This approach requires an offline optimization process and its performance is highly sensitive to model error, which prevents the proposed approach working in many real-time applications. The passive method on the other hand uses preloaded passive joints in order to eliminate backlash throughout a desired workspace when given norm-bounded external loads [8]. Although much simpler than the active method, the passive method is hard to realize on a parallel robot with more than three degrees of freedom and is not feasible with large external loads of the order of 100N or greater. 3

7 In this paper the authors investigate combining the benefits of both active and passive preload methods using actuation redundancy to prevent backlash on a six degree of freedom Stewart platform. Rather than using offline optimization based on feed-forward dynamics, an online optimization algorithm is developed combined with a feedback force control scheme to achieve a real-time method which is robust to both model inaccuracy and load disturbance. The proposed approach is ideal for applications where the Stewart platform is required to implement a high-precision task under large external loads, e.g. materials testing, machining, assembling, etc. Section 2 presents the backlash free condition, which is the essential goal for preload control. Based on the backlash free condition, the overall solution is formulated, followed by four main problems to be further treated in Section 3: the configuration of the redundant manipulator (Subsection 3.1), the inverse dynamics equation (Subsection 3.2), the preload optimization algorithm (Subsection 3.3), and the force control scheme on the redundant actuator (Subsection 3.4). Section 3.5 presents the simulation results on a custom-built Stewart platform-based manipulator developed for testing biological materials [9], followed by Section 3.6 which uses physical experiment to further verify the results. 2. Problem Statement A general Stewart platform mechanism consists of six linear actuators, which are connected via universal joints to a fixed base below and via spherical joints to a moving platform above. Ballscrews driven by rotary motors are often used as the linear actuators. The backlash in the ballscrew actuators is dominant compared to all other sources of backlash. The backlash-free condition for a linear actuator is shown in Fig. 1, 4

8 where represents the actuator control forces, represents the backlash-free threshold, and represents the actuator payload limit. The backlash-free condition physically means the magnitude of the actuator control force must remain above a certain level and its sign must remain fixed during the period of the task for backlash prevention (Muller, 2005, Wei and Simman, 2010). Its mathematical expression is:,,. (1) In order to prevent backlash on a Stewart platform, all of its six ballscrew actuators must satisfy the backlash-free condition. For achieving this, a new preload control method is proposed with a redundant linear actuator attached to the moving platform (Fig. 2). Overall, the concept is to use the redundant actuator to regulate the preloads on the original position-controlled ballscrews for the purpose of ensuring the control forces remain in the backlash-free region. As the solution is not unique, this forms an optimal force control problem in which the redundant actuator is required to generate minimum internal preloads to satisfy the backlash-free condition with lowest cost. Therefore, a preload optimization algorithm is used to search for the desired preload on the redundant actuator ( ) based on the backlash-free condition, the inverse dynamics equation, and the varying parameters (e.g. external forces and moments, and the endeffector trajectory). In series with the optimization algorithm, a feedback force control scheme is used to drive the redundant actuator to achieve the desired preload. A kinematics-based dual loop PID control scheme [9] is used to control the original six ballscrews for accurately positioning the robot end-effector. This requires six linear encoders mounted in parallel with the six ballscrews to measure their absolute lengths. The use of dual loop PID control not only ensures the accuracy of the positioning when 5

9 backlash is eliminated by preload control but also guarantees the stability of the plant in the case backlash is not eliminated effectively. 3. Theoretical Analysis Four main problems are treated in this section for achieving the proposed preload control method in practice. Firstly, the configuration of the redundant actuator in the Stewart platform is analysed for ease of control. Secondly, a simplified inverse dynamics equation is derived for the redundant manipulator configuration. Thirdly, an online optimization algorithm is proposed to determine the preload requirement on the redundant actuator in real-time. Finally, the force control on the redundant actuator is investigated for the purpose of accurate tracking and disturbance rejection. 3.1 Redundant Manipulator Configuration The redundant actuation of parallel robots has been widely studied due to the advantages of eliminating singularities, increasing manipulator stiffness, payload and acceleration, and reducing power consumption [10,11]. These aspects are not the focus of this study. Instead, the redundant actuation of the Stewart platform is used to regulate the preloads assigned on the ballscrews for the purpose of backlash prevention. From a controllability point of view, this is difficult as all six ballscrews must satisfy the backlash-free condition with the regulation from only one actuator. There is no doubt that the configuration of the redundant manipulator is fundamental to the success of the proposed method. 6

10 As a Stewart platform is symmetrical in its nominal configuration, an external preload along the centre axis of the manipulator (z-axis of the global coordinate system) can effectively create preloads on all six legs. Therefore, the redundant actuator is configured to align with the centre axis (Fig. 3). The top end of the actuator is connected via a spherical joint to a rigid support frame whose mounting point is on the centre axis while the bottom end is attached via a spherical joint to the centre of the moving platform. Although misalignment between the redundant actuator and the centre axis occurs during motion of the moving platform, effective preloads can still be achieved on all six legs within the envelope of motion of a typical Stewart platform. With a sufficiently long redundant actuator assembly, it is possible to apply all compressions or all tensions on the six legs, and therefore largely decrease the overall control difficulty. Moreover, a passive element (mass-spring-damper system) is introduced into the redundant actuator assembly to achieve a moderately compliant coupling with the Stewart platform. This inherently increases the disturbance rejection and force resolution of the system [12], and thus makes the implementation of force control much easier. A force sensor is attached to the redundant actuator to measure its preload. Details of parameter selection for the passive element will be discussed in Section In this study, the redundant leg is placed at the upper space of the robot assuming that the inner space is used for implementing tasks which is often the case in applications involving large interaction forces (e.g. material testing and machining). If the upper space of the robot is required for task implementation, the redundant leg can be placed inside the inner space. 3.2 Inverse Dynamics Equation 7

11 The inverse dynamics model of a general Stewart platform has been presented in detail in Do and Shahimpoor [13], Dasgupta and Mruthyunjaya [14], and Harib and Srinivasan [15]. In the proposed preload control method, the inverse dynamics is used to predict the actuator control forces for optimizing the preload on the redundant actuator. To ease the computational expense of optimization, a simplified inverse dynamics model is derived for the redundant manipulator with the following assumptions: 1) As the motion range of the Stewart platform is limited around its nominal pose in high precision and high load applications, the centre of gravity of each leg is always fixed at the point which is the equivalent centre of gravity when the leg is at its nominal length (mid stroke). 2) A universal joint is used at the stationary end of each leg (includes the redundant leg) and therefore there is no rotational movement about the longitudinal axis of the leg. 3) Friction is not considered. 4) Motor dynamics and actuator transmission system dynamics are not considered. Figure 4 shows the free-body diagram of one leg and the moving platform. Each leg (actuator) consists of a cylinder and a piston. As the moving platform and each leg are connected via a frictionless spherical joint, there is no moment but a single force exerted at which can be decomposed as a force along the longitudinal axis of the leg ( ) and a force normal to the longitudinal axis ( ). results from the actuator control force and is caused by the rotational dynamics of the leg. In order to solve, must be solved first. Considering the moments acting on ith leg about the rotation centre of the leg, Euler s equation gives: (2) where is the unit vector along the leg, is the leg length, is the distance between the leg rotation centre and the leg centre of gravity, is the leg mass, represents the 8

12 inertia tensor of leg, and are the angular velocity and the angular acceleration of the leg respectively, is the acceleration of the gravity centre of the leg, and is the gravitational vector. The global basis can be obtained via the following equation (3) where is the inertia tensor of the leg relative to the leg inertia coordinate system and remains as a constant, is the rotation matrix describing the orientation of relative to the global coordinate system {O}. By assuming there is no rotation moment about the longitudinal axis of the leg (i.e. [14], and ), the kinematics of the leg can be written as Eqs. (4)-(8), (4) where represents the end-effector position, represents the end-effector orientation, represents the position of the ith spherical joint in the end-effector coordinate system {o}, and represents the position of the ith universal joint. (5) where represents the elongation speed of leg i, and and represents the angular velocity and the linear velocity of the end-effector respectively. The angular velocity and acceleration of the leg respectively are given by:, (6) (7) where and represents the angular and linear acceleration of the end-effector respectively. The acceleration of the centre of gravity of leg i can be written as 9

13 . (8) By substituting Eqs. (3)-(8) into Eq. (2), can be solved. Then, considering the dynamics of the moving platform, Newton s equation for the moving platform gives, (9) where represents the moving platform mass, represents the external forces acting on the platform in the end-effector coordinate system, and represents the position vector of the gravity centre of the moving platform in the end-effector coordinate system. Considering the moments acting on the moving platform about, Euler s equation gives, (10) where represents the external moments acting on the platform in the end-effector coordinate system, represents the position vector of the external force exerting point in the end-effector coordinate system, and represents the inertia tensor of the moving platform and can be obtained via (11) where system is the inertia tensor of the moving platform relative to end-effector coordinate and remains as a constant. The dynamics equation of the moving platform can be written in matrix form: 10

14 (12) with,,,,,,,, where represents the kinematics Jacobian matrix of a general Stewart platform, represents the preloads on six original legs, represents the inertia matrix of the moving platform, represents the centrifugal and Coriolis terms of the moving platform, represents the gravity vector of the moving platform, represents the terms generated from the dynamics of the legs, represents the statics vector of the redundant leg, represents the axial preload on the redundant leg, and represents the external loads. In Eq. (12), if, the external loads, and the trajectory of the end-effector are known, then can be calculated. Finally, considering the dynamics of the actuator piston, the actuator control forces can be derived:, (13) 11

15 where is the mass of the actuator piston, and is the elongation acceleration of leg i which can be written as. (14) Although simplified, the inverse dynamics model of the redundant manipulator is still difficult to solve in real-time. The model can be further simplified by eliminating all the Coriolis and centrifugal terms. In applications where the motion of the manipulator is slow and external loads are large, the dynamics of the legs and pistons can be ignored and therefore the inverse dynamics model can be finally simplified as a closed form: (15) with,, where represents the control forces for the original six actuators, represents the gravity matrix of the pistons, and represents the gravity vector of all seven legs. 3.3 Preload Optimisation Optimisation Problem Formulation Even if assuming the trajectories of the end-effector and the external loads are known, the solution for in Eq. (15) is not unique in satisfying the backlash-free condition as described in Eq. (1). This problem can be solved by minimizing the total internal 12

16 preloads acting on the seven actuators. The lower the total internal preloads means the lower the energy consumption of the system and smaller the redundant actuator. Therefore, the backlash prevention optimal control problem can be formed as:, (16) where,, where L represents the 2-norm of the total internal preloads at time t, represents the maximum allowed preload on the redundant leg. As is the only unknown in L and all six need to satisfy the backlash prevention condition, Eq. (16) is a one-dimensional quadratic optimization problem subject to seven inequality constraints. Furthermore, there are possible combinations of signs of, each of which has to be considered independently in Eq. (16) and therefore the required computational time for solving Eq. (16) is enlarged 64 times. As mentioned in Section 3.1, with the redundant actuator configuration, it is easy to add all compression (positive preloads) or all tension (negative preloads) on the six original legs and thus it is far more feasible to cause all positive or all negative rather than the other cases. This reduces the possible combinations of signs from 64 to 2 and the optimal control problem is simplified as only two possible cases:, or, (17) 13

17 3.3.2 Optimisation Algorithm Problem (17) can be solved by offline optimization methods if a prescribed trajectory of the end-effector and a prescribed trajectory of the external loads are both given. However, in real-time applications, the external loads are caused by the interaction between the robot and environment, and thus the prescribed trajectory of external loads is generally unpredictable. This prevents the offline optimization methods from working in applications where the external loads are dominant. In order to address this issue, the authors developed an online optimization algorithm. This approach requires the external loads to be measured by a 6-DOF load sensor which normally exists in high interactive force applications. With the external load feedback, can be observed and predicted for determining at each discrete time. As the proposed optimization algorithm is based on online feedback measurement rather than offline processing, may slip into the backlash-free condition before a control decision is made. Furthermore, when tracking the determined on the redundant leg under force control, force tracking errors must appear and cause preload errors on the six original legs. This can also lead to stray into the backlash problem region. Therefore, the backlash free-condition in Eq. (17) is redefined for compensating the delays in measuring external loads and controlling :, or, (18) where represents a safety margin which narrows the original backlash-free condition. If Eq. (18) is satisfied, are in the safe zone, where not only satisfy the backlash-free condition but also are away from the backlash problem region. If Eq. (18) 14

18 is not satisfied, are in the danger zone, and are either very close to or already in the backlash problem region. With the definition of Eq. (18), a decision can be made before the backlash problem actually occurs. The flow chart of the proposed algorithm is shown in Fig. 5. At each discrete time, the external forces ( ) and moments are measured from the 6-DOF load cell. and represent the current desired end-effector pose and acceleration respectively. represents the preload requirement for the redundant leg calculated at the last discrete time. Using the inverse dynamics Eq. (15), the current control forces of the six position-controlled actuators can be approximated as well as the current total internal preloads index. Then are checked in Eq. (18). If are in the safe zone, is regulated within its range to minimize the total internal preloads index in the range of the safe zone. In order to guarantee the smoothness of and decrease the computation burden, only the two points ( ) around with a small increment d are considered. The total internal preloads index and for these two points are calculated and compared. The smaller one is then compared with the current index. If is smaller, the preload requirement at remains the same as. Otherwise, the control forces under the new preload ( ) are calculated via Eq. (15) and checked in Eq. (18). If are in the safe zone, is equal to or. If not, remains the same as. In the case when are in the danger zone, is regulated in its range to quickly move the control forces into the safe zone. This is achieved by iteratively searching along both positive and negative directions simultaneously from. At each iteration j, and are increased in their directions with an increment of d. Then the corresponding control 15

19 forces and are calculated via Eq. (15) and checked in Eq. (18). If none of and are in the safe zone, the next iteration starts. Otherwise, the iteration ends and the rest of the code simply ensures the discrete increment between and is below a maximum allowed value. 3.4 Force Control In order to achieve the optimised preload trajectory on the redundant leg, an accurate force control is required. This subsection investigates the redundant leg dynamics as well as the control algorithm for preload tracking Dynamics Model of the Redundant Leg Figure 6 shows a simplified schematic of the redundant leg, where the system is modelled as a linear three-mass system under two assumptions. Firstly, we assume the connection between motor and ballscrew piston is infinitely rigid compared to the massspring-damper (MSD) system. Secondly, by assuming the ballscrew backlash is infinitely small compared to the MSD system displacement, backlash non-linearity is ignored. Analysing the torque balance on the motor, we have: (19) where is the motor moment of inertia, is the viscous motor friction, is the motor rotational angle, is the motor driving torque, is the control force for driving the ballscrew piston, and is the torque to force ratio of the ballscrew. As the ballscrew piston and MSD cylinder are bolted together, the differential equation describing their dynamics can be written in the form of a collective mass, 16

20 , (20), where is the total mass of the ballscrew piston and MSD cylinder, is the viscous friction of the ballscrew piston, is the displacement of the ballscrew piston, the displacement of the MSD piston is equal to the displacement of the redundant leg length, and are the spring stiffness and damping of the MSD system respectively, and is the angle to displacement ratio (also known as lead) of the ballscrew. Analysing the force balance on the MSD piston, we have: (21) where is the mass of the MSD piston, is the preload on the redundant leg. With Eqs. (19) to (21), the block diagram of the redundant actuator dynamics can be found in Fig. 7. Clearly, the preload to be controlled is subject to the acceleration term and gravity term and damping term of the MSD piston, and the stiffness term of the MSD system. The acceleration term and gravity term of the MSD piston are the disturbances in as they are not controllable via. Therefore, the MSD piston mass is ideally made as small as possible to minimize such disturbances. The stiffness term of the MSD system is the major term in which can be controlled by regulating using a position control loop of the redundant actuator. The selection of an appropriate MSD system stiffness is critical. Very high can lead to low disturbance rejection. This physically means any disturbance or error in spring movement can lead to large force errors in. Conversely, a very low can decrease the bandwidth of force control if the actuator slew rate is limited. The stability of force control is subject to the MSD 17

21 damping term. There is a trade-off in selecting the MSD damping. A low can cause control instability, while a high can lead to a large time constant and therefore decrease the bandwidth Force Control Algorithm As is mainly governed by the relative displacement between the ballscrew piston and MSD piston, a position-based explicit control algorithm [12] is applied to control. Figure 8 shows the algorithm in the form of a block diagram, where superscript d represents the desired value and superscript s represents the real value. An outer force control loop is placed around an inner position control loop. The force loop calculates the desired relative displacement between the ballscrew piston and MSD piston for minimizing the force error between the desired force and the measured force. is derived from the preload optimization algorithm while is measured from the sensor. The absolute displacement of the leg length is calculated via inverse kinematics from the end-effector desired pose and is used to compensate the impact of the displacement of the MSD piston on. The sum of and gives the total desired displacement of the ballscrew piston, while the real displacement of the ballscrew piston is obtained from the motor rotary encoder. The internal position loop calculates the motor torque based on the displacement error between and for driving the ballscrew piston to achieve the desired displacement. A PID controller is applied to the position control loop. The force controller consists of a pure integral term and a low pass filter: 18

22 (22) where is an approximation of the spring stiffness, is the integral gain, and is the low pass filter time constant. Integral control is commonly used in position-based force control and yields good accuracy. is normally set as half of the position-loop bandwidth with the resulting bandwidth of the force loop half the position-loop bandwidth. With integral control, the force error is proportional to the desired velocity of the actuator and therefore any discontinuity in force error can result in discontinuity in actuator motion. In order to ensure smooth actuator motion, a low pass filter is used in series. 4. Numerical Simulation This section uses a custom-built Stewart platform-based manipulator as an example to verify the preload control method with the assumption that an additional leg consisting of a ballscrew (same as the original leg ballscrews) and a mass-spring-damper system is mounted at the top of the manipulator. Simulations are implemented on a high fidelity model of this system in the aspects of the redundant manipulator configuration, the preload optimization algorithm, and the force control strategy. 4.1 Preload Distribution Efficiency of the Redundant Manipulator This subsection assesses the proposed redundant manipulator configuration in terms of its efficiency to distribute active preloads on the six position-controlled legs. A comparison was undertaken between locating the redundant leg at the robot upper space and placing the redundant leg into the robot inner space. The geometrical parameters of 19

23 the original manipulator together with the assumed geometries of the redundant leg are shown in Tables 1 and 2, which respectively present the coordinates of the fixed universal joints and the coordinates of the moveable spherical joints of the seven legs in the global coordinate system and in the end-effector coordinate system. Since the manipulator used for simulation implements tasks in its inner space, the redundant leg was firstly assumed to be located at the robot upper space (as illusrated in Fig. 3) and its joint coordinate locations were selected considering the dimensions of the robot moving platform, the length of the ballscrew actuator and the length of the MDS system. Then the dimensions of the other case, where the redundant leg is placed into the robot inner space, were obtained by simply mirroring the upper space leg dimensions about the XY plane of the end-effector coordinate system when the robot is at its nominal central pose [0mm 0mm 490.7mm ]. With the mirrored leg dimensions, a more comparable result can be obtained between the two cases. Given the geometrical parameters shown in Tables 1 and 2, the active preloads distributed on the six original legs arising from the unit compressive preload of the redundant leg can be calculated. In order to obtain the overall distribution efficiency of the redundant manipulator within the workspace of the robot, such a relationship is quantified during the movement of the robot along each of the three translational axes and the three rotational axes about a virtual centre of rotation (the interaction point between the robot and the environment). For the case that the redundant leg is located at the upper space, the virtual centre of rotation is defined as [0mm 0mm -100mm] in the end-effector coordinate system. For the other case, the virtual centre of rotation is defined as [0mm 0mm 100mm]. 20

24 As a typical example, the preload distribution efficiency on the x-axis translation is shown in Fig. 9, where the left subfigure shows the case when the redundant leg is located at the upper space while the right subfigure shows the case when the redundant leg is in the inner space. The solid lines represent the preload ratio of forces between the th leg and the redundant leg and the pink dashed lines represent the boundaries of the low efficiency zone [ ]. Preload ratios outside this zone means that effective preload can be distributed to the corresponding leg, whilst ratios within the zone leads to low distribution efficiency, in which circumstance the preload on the corresponding leg is difficult to control since very small active preload can be assigned on the axial direction of the leg. The worst case scenario is when the six preload ratios do not have the same sign. When this occurs, the hypothesis in Eq. 17 that the redundant leg can cause all tensions or all compressions on the original six legs is no longer valid, which can consequently cause null solution in the preload control algorithm. From Fig. 9a (upper case), we can see that the preload ratios are outside the low effciency zone and have the same signs only when the robot motion on the x translation is restricted to mm, which is about its half range of motion on this axis ( mm). By contrast, Fig. 9b (inner case) shows that effective preloads can be assigned on all six positioncontrolled legs (magnitude of the ratios > 0.1) over the full motion range. The results on the other five degrees of freedom are listed in Table 3. Clearly, when the redundant leg is placed at the robot upper space, the motion range of the robot must be restricted to ensure an acceptable overall preload distribution efficiency. By contrast, placing the redundant leg into the robot inner space leads to a satisfied overall preload distribution efficiency over the full robot motion range. In the later sections, the redundant leg is placed at the robot upper space to verify the proposed preload control method in both 21

25 simulation and experiment since the original manipulator was designed to implement tasks in its inner space. 4.2 Preload Optimisation Algorithm The proposed preload optimization algorithm is assessed in this subsection under the following two assumptions. 1) The robot interacts with a stiff environment within its inner space and therefore undergoes large 6-DOF external loads. For simplicity, the environment is assumed to have a linear stiffness matrix with diagonal terms only. 2) As the simulated motion is slow and the resulting external loads are large, the end-effector acceleration term and the leg dynamics terms are negligible and thus are ignored in Eq. (15) during optimization. The geometrical and physical parameters required for preload optimization are listed in Table 4. The backlash-free condition (,, and ) for the robot ballscrews are estimated from experiments. The loop running the preload optimization algorithm has a loop rate of 100Hz. The initial preload on the redundant leg is defined as 100N (a positive value means compression) for initially moving the control forces of all the original ballscrews into the positive backlash-free region, such that all six control forces are positive. Simulations are implemented by deforming the environment about its centre of rotation (interaction point) in shear (x and y axis translation), axial loading (z axis translation), bending (x and y axis rotation), and torsion (z axis rotation). A sinusoidal waveform with +/-3mm (+/-10 degrees for rotation) amplitude and 0.1Hz is applied on each of the above six degrees of freedom sequentially for three cycles. In addition to the major movement, a sinusoidal waveform with +/-0.1mm amplitude and 0.1Hz is superposed to all three translational axes for 22

26 simulating the coupled forces arising from the movement of the environment centre of rotation. Figures 10 and 11 show the simulation results under x-axis shear and x-axis bending respectively, where subplot (a) shows the optimized control forces on the six positioncontrolled legs (solid lines) and the backlash-free threshold (pink dashed line) and subplot (b) shows the required preload on the redundant leg. The results demonstrate that the proposed algorithm is able to restrict the control forces on the six positioncontrolled legs to the backlash-free region by generating a consistent desired preload trajectory on the redundant leg. As we can further see from the plots, as soon as the control force on any of the six legs approaches the margin of the danger zone (defined as 80N in simulation), the algorithm enlarges the desired preload on the redundant leg in order to move the control forces on the six legs away from the backlash problem region. When the control forces on the six legs are in the safe zone, the algorithm gradually decreases the total internal preloads on all seven legs. Results also demonstrate that the proposed method is robust to large dynamic external loads. The cases on the other four degrees of freedom have similar results but are not shown here due to redundancy. 4.3 Force Control Algorithm The force control of the redundant leg was simulated in Matlab Simulink The redundant leg dynamics were modelled as the simplified system shown in Fig. 7, where the numerical values used for simulation are listed in Table 5. The parameters corresponding to the actuator dynamics (,,,, and ) were obtained from the Aerotech BM250 motor and EDRIVE VT actuator manuals. The parameters 23

27 corresponding to the MSD system dynamics (, and ) were selected to achieve a high bandwidth force control as well as good disturbance rejection. For example, the maximum backlash in the actuator is about 0.05mm which can only result in 5N disturbance with the selected spring stiffness. Therefore, ballscrew backlash of the redundant leg is negligible in simulation, as is the tracking error of the robot endeffector, which is normally within 0.05mm when the control forces on the positioncontrolled legs remain in the backlash-free region. The integral based force control algorithm shown in Fig. 8 runs in a 100Hz loop. The parameters of the force controller and the position controller in the equivalent continuous time domain are listed in Table 5. The simulation has been undertaken on the steepest preload trajectory (Fig. 10(b)) obtained above. As the desired preload on the redundant leg is periodic, only the section between the 11th second and 20th second which corresponds to the 2nd preload optimization cycle is displayed. Fig. 12 shows the simulated force tracking results. The maximum tracking error is approximately 45N. This physically means the maximum force error assigned on each of the six position-controlled legs is about 8N which is lower than the safety margin defined. Thus, a backlash-free condition is ensured even with a delay in the control so long as the safety margin is sufficiently large to tolerate the force control error. 5. Physical Experiment 24

28 The assembly of the custom-built redundant manipulator [9, 16] for experiment is shown in Fig. 13. The original manipulator consists of six EDRIVE VT actuators driven by Aerotech BM250 motors. An AMTI MC load cell is mounted on the top platform to measure the 6-DOF loads reacted from deforming the testing sample a stiff polymer specimen for this experiment. A pyramid shape support frame was designed to mount the redundant leg at the upper space. The framework was manufactured from powder coated RHS steel. Static and vibration analyses were implemented on the framework in ANASYS Workbench during the design process. The final design has a stiffness of about 80000N/mm on the compression/tension axis and a stiffness of about 14000N/mm on the shear axes. The first natural frequency of the framework is about 87Hz. A seventh EDRIVE VT actuator is used to drive the redundant leg. The piston of the actuator is coupled to a NET motorbike shock absorber, which acts as the MDS system. The motorbike shock absorber was chosen due to its availability, compact size and stiffness. The static performance of the shock absorber (Fig. 14) was directly measured using an Instron model 8511 material testing machine. The shock absorber exhibits a desired linear performance with a stiffness of about 30N/mm and a damping constant of about 8N/(mm/s) only under a compressive force between 150N and 1000N. This is not ideal in real applications but is sufficient for verifying the proposed concept. The shock absorber is then coupled to a Novatech F214 load stud. The location of the load stud allows direct measurement of the preload exerted on the manipulator top platform. Spherical joints are used on both ends of the redundant leg to couple the leg to the manipulator and the frame. Figure 15 shows the overall control hardware configuration for the preload control experiment. The control system of the original manipulator runs a host-target structure 25

29 [9]. A host computer runs Windows and LabVIEW graphical user interface for operating the system. Connected with the host computer via Ethernet, a target real-time controller (NI PXI 8106) is used to handle upper level control of the manipulator. At the lowest level, two FPGA boards (NI PXI 7852R) connect with the real-time controller via DMA and run the dual loop PID controllers for the six robot legs. The control signals are then sent to six Aerotech Soloist amplifiers (CP20) which drive the leg motors. An AMTI MSA-6 strain gauge amplifier converts the AMTI load-cell analog signal to a digital form and is sent serially over RS232 in order to minimize the noise arising from the motor servo amplifiers. The converted RS232 signal is then input into the real-time controller via a serial port on the controller and is decoded using the builtin NI VISA. In this way, the measured loads are obtained at a 200Hz sampling rate and the noise in the obtained signal is about ±6N and ±0.3Nm. For the same reason, a custom-built strain gauge amplifer is used to digitize the Novatech load stud signal and it is sent via a custom-written RS232 protocol on the FPGA. Unfortunately the obtained preload signal contains noise as high as ±70N which mainly arises from the large measurement range of the load stud (±15000N). This figure is far beyond the acceptable range for the experiment. To reduce the noise, a smaller force sensor with a lower capacity is required, however this would reduce the stiffness of the redundant leg and consequently degrade its dynamic performance. An alternative solution estimating the preload from the deformed displacement (travel) of the shock absorber is applied to avoid directly measuring the preload. The deformed displacement of the shock absorber can be obtained by comparing the difference between the robot travel pose and 7th leg travel pose. Then the linear function between the travel of the shock absorber and the 26

30 force response as shown in Fig. 14 can be used to estimate the preload on the redundant leg. The preload optimization and the force control algorithm shown in Figs. 5 and 8 run at 100Hz on the real-time controller. A Maxon EPOS2 70/10 position controller is used to run the inner position loop (as a form of velocity control at 10kHz loop rate) on the redundant leg shown in Fig. 8. As LabVIEW real-time controller does not support the Maxon LabVIEW driver (which only works under LabVIEW Windows), the velocity command of the motor on the 7th leg, which is calculated from the force control loop, has to be sent to the Maxon controller indirectly via the host PC at 50Hz sampling rate. A high density polymer specimen was mounted on the redundantly actuated manipulator to emulate the robot interacting with a stiff environment which undergoes large reactive external loads. Most of the control parameters for the experiment were defined the same as the values for the simulation, and where they differ are stated in Table 6. The backlash-free threshold and the safety margin were increased to 80N and 20N respectively to compensate the delay and error arising from the limitations of the control hardware set-up. The payload generated by the redundant leg was restricted between 150N and 1000N to ensure that the shock absorber remains within its linear range. For the redundant leg, the force control gains were selected by trial and error and position control gains were tuned by the Maxon EPOS2 controller auto-tuning system. The robot was commanded to deform the polymer specimen along each of the 6-DOF under two circumstances. In the first circumstance, the robot was controlled without the redundant leg but with a dead mass preload (180N) on top of the robot. Under the second circumstance, the robot was controlled with the redundant leg using the proposed active preload control method. To obtain comparable results, all the common parameters (e.g. control gains of the position-controlled legs) and testing protocols were 27

31 defined as the same for both circumstances. The testing protocols included shearing the specimen by 1mm along the x and y axis, compressing the specimen by 0.4mm along the z axis, bending the specimen by 6 degrees about the x and y axis, and twisting the specimen by 6 degrees about the z axis. For shearing, compression and torsion testing, the displacements were applied as a form of haver-sine waveform at 0.02Hz for three cycles. For bending testing, the displacements were applied as a form of haver-sine waveform at 0.01Hz for three cycles. These protocols were chosen for the following reasons. Firstly, the displacements were selected to ensure the resulting external loads on the robot were within the allowable range, which can be addressed by the force capacity (150N to 1000N) of the shock absorber. Secondly, the testing speed was defined in a very slow manner to minimize the error from preload estimation, where only the static force was considered and to also tolerate the delay in the control hardware set-up. Finally, backlash instabilities normally occurred at slow test speeds which meant that the actuators spent considerable time in the backlash region during zero crossings of actuator load. Furthermore a slow motion allowed the limit cycles arising from backlash instabilities to become dominant and obvious within the overall robot dynamic tracking inaccuracies. Figure 16 shows the experimental results under x axis shear, where subfigures (a) and (b) represent the three translational and three rotational errors of the robot respectively, and subfigure (c) represents the preloads on the six position-controlled legs. The plots on the left hand side illustrate the results under the dead mass preload method, while the plots on the right hand side illustrate the results under the active preload control method using the redundant leg. A maximum of 100N reactive shear resulted from the x axis shear testing. Under the dead mass preload method, the preloads on the position-controlled 28

32 legs were inevitably moved into the backlash-problem region as shown in subfigure (c). As soon as this happened, the stability margin of the corresponding leg was narrowed and consequently limit cycles arose from backlash instabilities as shown in subfigures (a) and (b). Such high frequency limit cycles can be harmful to the ballscrews and other mechanical components of the robot. By contrast, under the active preload control method, the preloads on all six position-controlled legs were consistently kept in the backlash-free region as shown in subfigure (c). Under such a condition, the non-linear dynamics of the backlash was eliminated in the leg dynamics and consequently backlash instabilities disappeared as shown in subfigure (a) and (b). As a result, the robot tracking abilities were significantly improved. The experiments on the other five degrees of freedom have similar results. The RMS tracking errors of the robot for the dead mass preload (DMP) method and for the active preload control (APC) method on each of the six degrees of freedom testings were computed and compared in Table 7. The RMS errors arising from APC were within 5µm on translational axes and 5 arcsecond on rotational axes which are about 2 to 15 times smaller than the counterparts arising from DMP. This proved the efficacy of the proposed active preload control method. 6. Discussion and Conclusion This paper studied the use of actuation redundancy to eliminate backlash inaccuracy for a general 6-DOF Stewart platform. A novel redundancy arrangement with a refined active preload control method was proposed for real-time control applications. Simulation results demonstrated that placing the redundant leg into the robot inner space results in a more effective preload distribution efficiency of the redundant 29

33 manipulator within its workspace compared to placing the redundant leg at the robot upper space, particularly along the horizontal (shear) axes of the robot. Thus, it is suggested to apply the inner space case in applications which require use of the robot's full range of motion (e.g. machining, assembling). Simulation results also demonstrated that the proposed real-time preload control algorithm can effectively achieve backlashfree conditions of the robot under large dynamically varying external loads. Because of the hardware limitations, the experiment was restricted to low speed tests, however, based on simulation results, it is expected that using improved hardware, the bandwidth of testing could increase. The experimental results further demonstrated that the proposed method can significantly mitigate (or even completely eliminate with an improved design) backlash instabilities from control and consequently higher bandwidth control can be achieved on the robot with higher accuracy compared to the same system without the redundant leg. In order to make the proposed active preload method fully applicable in industry, further design and research work is required. Firstly, the design of the redundant leg assembly is critical. A bicycle shock absorber is not ideal, not only because of the unsatisfactory dynamic performance on its longitudinal axis but also due to the unexpected dynamic behaviour on its transversal axes. Thus, a more sophisticated mass-damper-spring system needs to be designed to allow a single degree of freedom linear compliant motion along its longitudinal axis only. As the redundant leg actively controls the preloads on all six position-controlled legs, the load capacity of the redundant leg is required to be approximately 4 times higher than the position-controlled leg to ensure the controllability of the system. 30

34 References [1] Khalil, I.S.M., Golubovic, E., and Sabanovic, A., 2011, High precision motion control of parallel robots with imperfections and manufacturing tolerances, In Proc International Conference on Mechatronics ( ICM), Istanbul, Turkey, pp [2] Briot, S., and Bonev, I.A., 2008, Accuracy analysis of 3-dof planar parallel robots, Mechanism and Machine Theory 43, [3] Nordin, M., and Gutman, P.-O., 2002, Controlling mechanical systems with backlash a survey, Automatica 38, [4] McInroy, J.E., 2002, Modeling and design of flexure jointed Stewart platforms for control purposes, IEEE Transactions on Mechatronics 7(1), [5] Kang, B.H., Wen, J.T.-Y., Dagalakis, N.G., and Gorman, J.J., 2005, Analysis and design of parallel mechanisms with flexure joints, IEEE Transaction on Robotics 21(6), [6] Muller, A., 2005, Internal preload control of redundantly actuated parallel manipulators Its application to backlash avoiding control, IEEE Transactions on Robotics 21(4), [7] Boudreau, R., Mao, X., and Podhorodeski, R., 2011, Backlash elimination in parallel manipulators using actuation redundancy, Robotica 30, [8] Wei, W. and Simaan, N., 2010, Design of planar parallel robots with preloaded flexures for guaranteed backlash prevention, ASME Journal of Mechanisms and Robotics 2, (1-10). 31

35 [9] Ding, B., Stanley, R.M., Cazzolato, B.S., and Costi, J.J., 2011, Real-time FPGA control of a hexapod robot for 6-DOF biomechanical testing, In Proc. 37 th Conference of the IEEE Industrial Electronics Society (IECON), Melbourne, Australia, pp [10] Wang, H., Zhang, B.J., Liu, X.Z., Luo, D.Z., and Zhong, S.B., 2011, Singularity elimination of Stewart parallel manipulator based on redundant actuation, Advanced Materials Research , [11] Nahon, M.A., and Angles, J., 1989, Force optimization in redundantly-actuated closed kinematics chains, in Proc. IEEE International Conference of Robotics Automation (ICRA), Scottsdale, AZ, USA, pp [12] De Schutter, J., and Brussel, H.V., 1988, Compliant robot motion II. A control approach based on external control loops, International Journal of Robotics Research 7(44), [13] Do, W.Q.D., and Shahimpoor, M., 1998, Inverse dynamics analysis and simulation of a platform type of robot, Journal of Robotic Systems 5(3), [14] Dasgupta, B., and Mruthyunjaya, T.S., 1998, The Stewart platform manipulator: a review, Mechanism Machine Theory 35, [15] Harib, K. and Srinivasan, K., 2003, Kinematic and dynamic analysis of Stewart platform-based machine tool structures, Robotica 21(5), [16] Ding. B., Cazzolato, B.S., Grainger, S., Stanley, R.M., and Costi, J.J., 2013, Active preload control of a redundantly actuated Stewart platform for backlash prevention, In Proc IEEE Conference on Robotics and Automation (ICRA), Karlscrhe, Germany. 32

36 Figure Captions Fig. 1 Backlash-free condition for a linear actuator Fig. 2 Schematics showing the preload control method where and represents the external forces and moments,,, and represents the end-effector trajectory, velocity, and acceleration, represents the desired preload on the redundant actuator, and and represent the control forces for driving the redundant actuator and the original six ballscrews respectively. Fig. 3 Configuration of the redundant manipulator where BSP represents the ballscrew piston, M1 and M2 represent the upper mass and bottom mass of the mass-springdamper system respectively, FS represents the force sensor, and SJ represents the lower spherical joint. Fig. 4 Free-body diagram of one leg and the moving platform, where represents the fixed joint centre of leg i, represents the moving joint centre of leg i, represents the gravity centre of leg i, represents the end-effector, represents the gravity centre of the moving platform, and represents the point where the external loads are exerted on the moving platform. The end-effector coordinate system frame {o} is attached to o, a leg inertia coordinate system frame is attached to and rotates in coincidence with leg i, and a global coordinate system frame {O} is fixed for reference. (i=1:7). Fig. 5 Online optimization algorithm at discrete time. 33

37 Fig. 6 Simplified schematic diagram of the redundant actuator. Fig. 7 Block diagram of the redundant actuator dynamics. Fig. 8 Block diagram of position-based force control. Fig. 9 Preload distribution efficiency on x-axis translation. Fig. 10 Optimized control forces and desired preload under N x-axis shear. Fig. 11 Optimized control forces and desired preload under Nm x-axis bending. Fig. 12 Simulated force control performance on the redundant leg. Fig. 13 Assembly of the redundantly actuated manipulator experimental rig. Fig. 14 Measured static response of the shock absorber (tested under displacement control with a haversine waveform of a -35mm amplitude at 0.01Hz for three cycles). Fig. 15 Schematics showing the control hardware configuration for the preload control experiment and the communication between the hardware elements. Fig. 16 Comparison between dead mass preload (left figures) and active preload control using the redundant leg (right figures). The robot was commanded to shear the 34

38 specimen by 1mm along x-axis using a haver-sine waveform at 0.02Hz for three cycles. Maximum shear force reached 100N. 35

39 Table Captions Table 1 Coordinates of the fixed universal joints in the global coordinate system {O}. Table 2 Coordinates of the movable spherical joints in the end-effector coordinate system {o}. Table 3 Preload distribution efficiency of the redundant manipulator on each of the six degrees of freedom. Table 4 Geometrical and physical parameters for preload optimization simulation. Table 5 Model parameters for force control simulation. Table 6 Control parameters for physical experiment (same as the simulation parameter if not listed). Table 7 A comparison between the RMS tracking errors of the robot under dead mass preload and under active preload control. 36

40 Table 1 Table 1. Coordinates of the fixed universal joints in the global coordinate system {O} (upper space) (inner space) X (mm) Y (mm Z (mm)

41 Table 2 Table 2. Coordinates of the movable spherical joints in the end-effector coordinate system {o} (upper space) (inner space) X (mm) Y (mm Z (mm)

42 Table 3 Table 3. Preload distribution efficiency of the redundant manipulator on each of the six degrees of freedom Full Range of Motion Restricted Motion Range Lowest Preload Ratio (absolute) Upper Case Inner Case Upper Case Inner Case X axis translation mm mm N/A Y axis translation mm mm N/A Z axis translation mm N/A N/A X axis rotation N/A Y axis rotation N/A Z axis rotation N/A N/A

43 Table 4 Table 4. Geometrical and physical parameters for preload optimization simulation Parameters Values Description (units) Linear stiffness of the environment (N/mm, Nm/degree) Position of the platform centre of gravity in {o} (mm) Position of the interaction point in {o} (mm) 240 Length between leg rotation centre and gravity centre (mm) 20 Platform mass (kg) 2 Actuator piston mass (kg) 5 Leg mass (kg) 70 Backlash-free threshold (N) 4000 Actuator payload limit (N) 4000 Redundant actuator payload limit (N) 10 Safety margin (N) 2 Preload searching resolution (N) 20 Preload discrete increment limit (N) 100 Initial preload on the additional leg (N)

44 Table 5 Table 5. Model parameters for force control simulation Parameters Values Description (units) Lead of the ballscrew actuator (mm/rad) 2215 Torque to force ratio of the ballscrew actuator (N/Nm) Motor moment of inertia ( ) Motor viscous friction ( ) 5 Viscous friction of the ballscrew piston ( ) 2 Total mass of the ballscrew piston and MSD cylinder (kg) 1 Mass of the MSD piston (kg) 5 Damping coefficient of the MSD system ( ) 100 Spring stiffness of the MSD system ( ) 1.2 Proportional gain of the position PID controller ( 2.4 Integral gain of the position PID controller ( ) Derivative gain of the position PID controller ( ) 10 Integral gain of the force controller (1/s) 0.01 Low pass filter time constant of the force controller

45 Table 6 Table 6. Control parameters for physical experiment (same as the simulation parameter if not listed) Parameters Values Description (units) Position of the specimen centre of rotation in {o} (mm) 80 Backlash-free threshold (N) 20 Safety margin (N) 1000 Redundant actuator payload upper limit (N) 150 Redundant actuator payload lower limit (N) 30 Spring stiffness of the shock absorber ( ) 2 Integral gain of the force controller ( ) Low pass filter time constant of the force controller

46 Table 7 Table 7. A comparison between the RMS tracking errors of the robot under dead mass preload and under active preload control ffffmethod Axis ffffff Shear (x-axis) Shear (y-axis) Compression (z-axis) Bending (x-axis) Bending (y-axis) Torsion (z-axis) DMP APC DMP APC DMP APC DMP APC DMP APC DMP APC Tx ( m) Ty ( m) Tz ( m) Rx ( ) Ry ( ) Rz ( ) Maximum Load 100N 120N 500N 45Nm 45Nm 27Nm

47 Figure 1 Backlash free With backlash Backlash free

48 Figure 2 Preload optimization based on backlash- free condition and inverse dynamics Feedback force control Kinematics based dualloop PID control

49 Figure 3 Support Frame Redundant Actuator BSP M1 x z y M2 FS SJ

50 Figure 4 o {o} { } {O}

51 Figure 5 Trajectory generation BEGIN Read loads Initialization Inverse dynamics Minimize Y Check if in N Move into safe zone the safe zone Calculate Calculate Y Check if < N Calculate Calculate N N Check if < Check if < Calculate Y Y Calculate Check if in the safe zone Y N Check if in the safe zone Y N Check if in the safe zone N N Check if in the safe zone Y Check if Check if Y Y Y N N END

52 Figure 6 Motor Ballscrew piston and MSD cylinder MSD piston

53 Figure

54 Figure 8 + Inverse kinematics Force controller + + Position controller Fig. 7 Motor Motor encoder Force sensor

55 Figure 9 (color on web) Preload ratio (fui/fu7) Preload ratio (fui/fu7) st leg 2nd leg 3rd leg 4th leg 5th leg 6th leg LE line st leg 2nd leg 3rd leg 4th leg 5th leg 6th leg LE line Robot displacement along x-axis translation (mm) Robot displacement along x-axis translation (mm) a) Redundant leg at the upper space b) Redundant leg at the inner space

56 Figure 10 (color on web) Actuator control force (N) Preload (N) st leg 2nd leg 3rd leg 4th leg 5th leg 6th leg BFT th leg Time (seconds) (a) Optimized actuator control forces on the six legs Time (seconds) (b) Desired preload on the redundant leg

57 Figure 11 (color on web) Actuator control force (N) Preload (N) st leg 2nd leg 3rd leg 4th leg 5th leg 6th leg BFT th leg Time (seconds) (a) Optimized actuator control forces on the six legs Time (seconds) (b) Desired preload on the redundant leg

58 Figure 12 (color on web) Force (N) Actual force Desired force Time (seconds) (a) Force tracking performance in a cycle Actual force Desired force Time (seconds) (b) Zoomed-in section showing the maximum error

59 Figure 13 (color on web) Ballscrew Actuator & Motor Support Frame 6-DOF Load-cell Shock Absorber Force Sensor Polymer Specimen