A Modular Controller for Redundantly Actuated Tendon-Based Stewart Platforms

Transcription

1 Proceedings of EuCoMeS, the first European Conference on Mechanism Science Obergurgl (Austria), February Manfred Husty and Hans-Peter Schröcker, editors A Modular Controller for Redundantly Actuated Tendon-Based Stewart Platforms Tobias Bruckmann Andreas Pott Daniel Franitza Manfred Hiller Tendon-based Stewart platforms are capable of high speeds and accelerations. Thus, a reliable controller running with a high frequency is required. Here, an implementation based on a modular controller architecture is shown. To control the platform on a given trajectory, the implemented position control has to be extended by a tendon force control which generates defined and safe force values. In the case of one redundant tendon, a direct calculation is possible. Having at least two redundant tendons, the computation of a force distribution requires optimization methods, which are numerically expensive. In this paper, algorithms for both cases and their embedding into the controller are presented. 1 Introduction At the Chair of Mechatronics, the teststand for tendon-based Steward-platforms Segesta 1 has been developed during the past few years. Presently, the Segesta teststand uses seven (and - in a future modified version of Segesta - eight) tendons to move the platform on desired trajectories. To control the platform position, the actual distances between the winch points on the teststand frame and the tendon fixation points on the platform are required. These distances are measured by angular encoders at the winches which measure the tendon lengths with a resolution of approximately 0.05mm. Currently the platform is guided using position control in the domain of tendon lengths. Following a trajectory, intermediate poses for every time step are calculated. For these points, the inverse kinematics delivers the corresponding tendon lengths. Since the actual tendon lengths are available, feedback control is used to guide the platform. This basic control concept provides satisfying results for simple demonstrations of Segesta s capabilities. Looking at the precision of the movement, it was found out that the platform begins to wobble at higher speeds and Chair of Mechatronics, University Essen-Duisburg, Germany, {bruckmann,pott,hiller}@imech.de 1 Segesta - Seilgetriebene Stewart-Plattformen in Theorie und Anwendung 1

2 EuCoMeS 1st European Conference on Mechanism Science Figure 1: Segesta Testbed accelerations. This happens simultaneously to slack tendons. To prevent slackness and also to limit forces to the tendon s breaking load, tendon forces have to be controlled and one has to include lower and upper bounds. The calculation of a force distribution is straight forward in the case of a manipulator with m = n + 1 tendons, while in the case of m > n + 1, optimization is required, which is always expensive in terms of computation time. Since Segesta is also designed to work with online control, computation of the control values must be numerically efficient (or even fulfill realtime requirements, if possible), which is subject of elapsed and present work. The paper is structured as follows: In chapter 2, a short description of kinematics and dynamics is given and in chapter 3, methods for generating continuous force distributions are proposed. An example illustrates the resulting force distributions. The implementation for the use within the controller structure is described. Chapter 4 describes the hard- and software structure of the Segesta teststand with the focus on the used modular controller architecture. Finally, in chapter 5, the conclusions are drawn and an outlook is given. 2 Kinematics and Dynamics Segesta consists of two main components: a frame of aluminium profile bars which carries motors, winches as well as further components like computers, measurement equipment etc. 2

3 T. Bruckmann et al.: Modular Controller... (see chapter 4). The triangular platform is connected to the winches by tendons. Segesta is designed as an reconfigurable system by using modules which carry winches and motors and which can be installed and removed easily. Due to its lightweight structure, Segesta can generate high-dynamic motions. Along a trajectory, platform poses are calculated. Having the platform pose for every step, computation of the tendon lengths (inverse kinematics) is trivial compared to the in the general case complicated forward kinematics. Figure 2: Symbol Definitions for a General Tendon-Based Stewart-Platform Segesta can be described using the following vectors and coordinate frames, with µ = 1,..., m (Fang, 2005): The coordinate frame C B is the base frame, while C P is connected to the platform. The vectors b µ denote the positions of the winch points, i.e. the points where tendons are led through small ceramic eyes which are fixed. p µ are the vectors of the connecting points on the platform with respect to the platformfixed coordinate system C P. l µ denote the tendon vectors from the platform to the winches. The forces in the tendons are described by f µ, where f P and τ P denote all other applied forces and torques acting on the platform. Since tendons can only transmit pulling forces, tensions must always be greater than zero which leads to the need for at least m = n + 1 tendons (where n is the number of d.o.f. s and m the number of tendons) when no external load is available to tighten the tendons (Husty et al., 3

4 EuCoMeS 1st European Conference on Mechanism Science 2002). The force equilibrium for the platform can be easily expressed as (Ming and Higuchi, 1994) where [ ν 1... ν m p 1 ν 1 p m ν m ] f 1. f m + [ fp τ p ] = 0, f > 0 (1) ν = l µ l µ (2) or in a more compact form as A T f + w = 0. (3) 3 Force Optimization Force control is necessary to guarantee a defined tension distribution. Thus, a method to calculate tendon forces must be provided. Because we have force redundancy in the examined systems with m n+1 and thus at least a one-dimensional solution set for the force distributions belonging to a specific position, an optimal solution is desired. The term optimal has to be used with respect to the technical application (Verhoeven, 2003). In this paper, optimal means small forces, but it is also possible to generate high forces (higher eigenfrequency and stiffness) or force distributions in between (security and robustness against parameter changes). Beside minimum tendon forces (which can be zero as smallest possible force) also maximum tendon force are of great importance since their ratio defines the workspace borders. To evaluate the proximity of a specific position of the platform to workspace borders, knowledge of the tendon forces is presumed. Obtaining a solution from the optimization algorithm which exceeds the tendon force limits means that the platform is outside the predefined workspace. So, the calculation of force distributions plays also a role in terms of reliability and security. In practice, it is of great importance to find continuous solutions. Non-continuous tendon forces may consist of acceptable solutions, but since those values are needed for control, they would cause steps in motor torques which leads to vibrations and high mechanical loads. 3.1 Single Redundancy The structure matrix A T has the dimension n m, which reflects the fact of under-determinateness. To solve (3) with respect to f, a least square algorithm (pseudo inverse) is used f = A }{{ +T w } + }{{} hλ, (4) f lsq f krn where A +T is the pseudo inverse of A T and h is the m 1-dimensional kernel of A T (Verhoeven, 2003). The kernel (or nullspace) h of A T is defined by 4

5 T. Bruckmann et al.: Modular Controller... A T h = 0, (5) which means that while it applies internal tension, it has no influence on the platform load. Figure 3: Force Distributions for three Tendons (Verhoeven, 2003) The solutions for f can be visualized for a more simple case of a planar manipulator with three tendons (fig. 3). In this case, the limits for minimum tendon force f min and maximum tendon force f max form a cube. All solutions for f can be represented by a line. If solutions exist, the cube and the line have an intersection which contains all acceptable solutions, i.e. solutions where no tendon exceeds its minimum or maximum force limits. The term f lsq = A +T w is a least square solution, that lies somewhere on the line, but not necessarily within the cube which means that it is not acceptable in this case. On the other hand, the term f krn = hλ provides the possibility to move the solution on the line since it only tenses or reliefs the tendons without violating the force equibrilium. In case of single redundancy the kernel has the dimension m 1, which means that we have one d.o.f. to move along the line via choosing a suitable λ. Based on this, a simple algorithm can be provided: 1. Calculate the structure matrix A T based on a given platform pose. 2. Compute the kernel h of the structure matrix. 3. Get the least square solution. 4. Calculate the force offset f offµ which has to be added to each tendon, which exceeds its lower bounds in the least squares solution. f off = f min f lsq (6) 5. Compute the necessary kernel multiplication factors λ µ for these tendons to achieve the force offset via kernel addition. λ µ = f offµ h µ (7) 5

6 EuCoMeS 1st European Conference on Mechanism Science 6. Select the maximum multiplication factor λ µmax. 7. Correct tendon forces using kernel h and λ µmax. 8. Check if no tendon is exceeding its bounds. If at least one tendon does, there is no solution. The algorithm has been implemented using M BILE and LAPACK (version 3.0) routines. The function DGESVD (providing singular value decomposition, SVD) was used to calculate the kernel and the function DGELSD (computes the minimum-norm solution to a real linear least squares problem) was chosen to get the least squares solution. 3.2 Multiple Redundancy In the case of m > n + 1, the tendon force distribution is generated with the m (m n)- dimensional kernel of A T. A Segesta teststand having eight drives is an example of this category. Here, in (4) the vector h is replaced by a matrix H and the scalar λ becomes the vector λ. f = A +T w + Hλ (8) This leads to a more complex optimization problem, since now there are (m n) redundant d.o.f. s. Again, it is relatively easy to find acceptable solutions, but it needs some consideration to find continuous solutions along a trajectory. One could try to generate continuous force distributions by defining a constant force for (m n) tendons and calculating the forces for the other n tendons, but this leads to an artificially reduced workspace. To apply this method, one has to select the tendons which have a predefined tension. Along a trajectory, it can be necessary to exchange some tendons between having a predefined or a calculated force. This happens, if the predefined selection results in a force distribution, violating the force bounds and holds the danger of non-continuous force distributions. Thus, this method is not usable for control purposes. Since this approach does not provide acceptable solutions, another algorithm must be found, to guarantee continuous tendon forces. In the domain of optimization, continuity depends on the cost function, which is given to the optimizer. Intuitive ideas like choosing a maximum (infinity) norm v = max 1 µ m f µ (9) result in non-continuous force distribution runs. A helpful choice is the p-norm m f p = p fµ, p (10) as proven by (Verhoeven and Hiller, 2002). In the current implementation, p = 2 was chosen which fits well to common optimizer implementations, using quadratic cost functions. This leads to an optimization problem with (m n) unbounded parameters λ α, 1 α (m n), and (2 m) linear constraints beside one non-linear (possibly square) cost function. Thus, it is possible, to get the equations for the optimizer. µ=1 6

7 T. Bruckmann et al.: Modular Controller... Parameters: Linear Constraints Cost Function f 2 = λ α f µoff m n ker µ,α λ α f µmax f µlsq α=1 m fµ 2 µ=1 Again, the algorithm has been implemented, using the mentioned M BILE and LAPACK routines. For optimization, the NAG library offers a variety of routines. The routine nag_opt_lin_lsq (solves linearly constrained linear least-squares problems) was chosen, since it fits best to this kind of optimization problems. As an example, a tendon-based Stewart platform with the frame dimensions [mm] and eight tendons has to follow a straight line trajectory which starts at 100mm above the ground and lifts the platform for 500mm. The platform orientation is constant with the Cardan angles ϕ = θ = ψ = 2.0. The minimum force desired is 2N, the maximum allowed force is 50N. The resulting force distribution is shown in fig. 4. It is continuous and remains between the given force bounds. Thus, it is feasible for control purposes. Figure 4: Force Distribution For Eight Tendons. The time history of the design variables in fig. 5 is of some interest: Though they are noncontinuous, they generate a continuous and therefore usable force distribution. Since kernel computation is done by SVD, the orientation of the vectors spanning the solution plane is variable and may change non-continuously. When a rotation of the spanning vectors occurs, the corresponding λ µ will also change, which leads to the described phenomenon. Considering computation time, this method shows a drawback. The calculation of the leastsquare solution and the kernel requires more time compared to the optimization itself. An alternative idea is to use the optimizer with a different set of linear constraints. Having the 7

8 EuCoMeS 1st European Conference on Mechanism Science Figure 5: Parameters λ 1, λ 2 For Eight Tendons structure matrix, it is also possible to get constraints which are different from the previous ones: Parameters: f µmin f µ f µmax Linear Constraints w µ = m A T α,µ f µ µ=1 m Cost Function f 2 = fµ 2 µ=1 Although this alternative approach has to deal with m instead of (m n) parameters, it was found out, that it is normally faster in computation time due to the good initial values, which can be reused from the last time step. Since on a trajectory the steps between two poses are small, the structure matrix and changes only a little bit. On the other hand, the kernel-based optimization using only (m n) parameters suffers from possible rotation of the vectors spanning the kernel. This effect arises from the SVD algorithm and makes the usage of the parameters from the previous step difficult. So for implementation in online control, the alternative approach using m parameters was chosen. Note that this iterative method is relatively fast, but it has no predictable worst case runtime. 4 Controller Structure Due to the characteristics of Segesta, as mentioned in section 2, a flexible controller is required allowing to implement the algorihms described in the previous section. When selecting a hardware/software combination, the following points have also to be taken into account: High Controller Frequency. Since Segesta can generate high-dynamic motions, it requires a high controller frequency (approx. 1000Hz sampling rate) combined with a good reliability. A realtime-capable solution for control is desirable. 8

9 T. Bruckmann et al.: Modular Controller... Figure 6: Combined Force- and Position Control Computing Power. A part of the numerical methods inside the controller requires a high computing power. Since some methods are iterative and thus not predictable in computation time, the option to distribute some tasks on different computers running asynchronously is needed. Man-Machine-Interface. Since Segesta s software structure is continuously being modified during further development, comfortable and powerful tracing and monitoring tools are required. To interact with the operator, a graphical user interface is desired. Availability of Numerical Libraries. Besides high computing power, numerical methods need an efficient and reliable implementation. So, the availability of numerical libraries (NAG, LAPACK, M BILE etc.) is important which saves time during development. Costs. Since Segesta s mechanical components can be manufactured at a relatively low price (especially in case of future usage within application), the controller should not rise the costs. 9

10 EuCoMeS 1st European Conference on Mechanism Science 4.1 Hardware Taking these aspects into account, a PC with a 1.3Ghz AMD Athlon CPU was chosen for the low-level control. It has two PEAK PK 100-X CAN Bus Cards controlling the winch drives and a ME2600 Analog Input Card for measuring the tendon forces via strain gages. For more complex methods (high-level control) an additional standard PC (Pentium IV/Athlon X2) providing more computation power is planned to do all expensive computational calculation for the low-level control PC. 4.2 Software Figure 7: Simple Graphical User Interface in MCA2 MCA2 (Modular Controller Architecture) (Forschungszentrum Informatik (Karlsruhe), 2004) is used as controller software since it meets the above mentioned requirements. MCA2 is developed by the Forschungszentrum Informatik in Karlsruhe, Germany, and is published under the GPL (General Public License) which allows a free usage. It is available for MS Windows and Linux. Moreover, it is capable of hard realtime using the Linux realtime extension RTAI (Realtime Application Interface)(The RTAI Project, 2004). Both Linux and RTAI are free of charges. MCA2 allows to create a modular controller structure. This means, the controller consists of different modules each having a defined task e.g. hardware-i/o, kinematics, sensor data filtering 10

11 T. Bruckmann et al.: Modular Controller... etc.. Interfaces between modules are well-defined by the number of in- and outputs enabling for simple exchange and modification of the modules. The controller is then assembled by coupling the modules and can be modified and expanded via removing, changing or adding modules. Moreover, MCA2 has a built-in GUI and online administration, allows realtime- and non-realtime tasks and even their mixed coupling. Distributed computation is one of the basic ideas of MCA2 and is transparent to the user, i.e. it makes no difference for module implementation, if the full controller is running on one or more PCs. 4.3 Modular Implementation Using both the results from the inverse kinematics and the force optimization makes a combined position-force-control possible, as shown in fig. 6. The position part delivers positioning precision while the force controller is responsible for positive tensions and acts as a kind of pilot control, since every force distribution belongs to a specific position at a given external load. Most of the modules working in the controller system are implemented based on explicit methods. Thus, a basic position control is possible. Since presently no realtime implementation of the discussed optimization methods is available, for advanced control a hybrid implementation was chosen which is fully supported by MCA2. While the basic control system is limited to methods allowing for realtime control, but provides guaranteed and constant reaction times, the additional force control is running in standard Linux, since a realtime system does not manage algorithms possibly exceeding the controller loop time. Moreover, in standard Linux a wide variety of common numerical libraries (e.g. NAG, LAPACK) can be adressed witout violating realtime conditions. If the force optimization exceeds its controller loop time, the basic realtime controller will detect this and either reuses the previous force values, disables force control or stops the system. In case a hard-realtime capable force optimization method can be found, the whole system can be switched to realtime providing a high reliability and safety. 5 Conclusions and Outlook In this paper, algorithms for generating continuous force distributions for the tendon-based Stewart-platform Segesta were presented. The implementation of these methods provides continuous force distributions. Furthermore, it is possible to set parameters, which allow the definition of the individual optimal result. Methods for the cases of m = n + 1 and m > n + 1 tendons, respectively, were presented. In the latter case, optimization algorithms have to be used, which are expensive in terms of computation time. Tests have shown, that the calculation of the required least square solution and the kernel is relatively time consuming. Thus, a formulation which does not need these precalculations was found, which is more suitable for online control. A modular controller structure implementing the described methods was shown. Presently, alternative algorithms using interval analysis are developed. It will be part of further investigations to compare both methods with respect to reliability, flexibility, availability for different operating systems and computation time. Special attention will be paid on their usability for realtime control (predictable computation time). Deriving realtime-capable algorithms from the methods stated here and embedding them into the described controller framework will be part of future research. 11

12 EuCoMeS 1st European Conference on Mechanism Science At the moment, the Segesta teststand is being equipped with strain gages. Simultaneously, a powerful PC will be set up and integrated into the MCA2-based controller structure, to perform numerically expensive computations as they were presented here. As soon as both are available, the combination of force- and position control can be realized and tested, and the enhancements regarding safety and precision of movement can be verified. 6 Acknowledgements This work is supported by the German Research Council (Deutsche Forschungsgemeinschaft) under HI370/24-1. References S. Fang. Design, Modeling and Motion Control of Tendon-Based Parallel Manipulators. Ph. D. dissertation, Gerhard-Mercator-University, Duisburg, Germany, Forschungszentrum Informatik (Karlsruhe). Modular Controller Architecture (MCA2) v4.0pre M. Husty, S. Mielczarek, and M. Hiller. A Redundant Spatial Stewart-Gough Platform with a Maximal Forward Kinematics Solution Set. Proceedings of the ARK 02, 8th. International Symposium on Advances in Robot Kinematics, A. Ming and T. Higuchi. Study on Multiple Degree of Freedom Positioning Mechanisms Using Wires, Part 1. Int. J. Japan Soc. Prec. Eng., 28: , The RTAI Project. Realtime Application Interface (RTAI) v R. Verhoeven. Analysis of the Workspace of Tendon-Based Stewart-Platforms. Ph. D. dissertation, Gerhard-Mercator-University, Duisburg, Germany, R. Verhoeven and M. Hiller. Tension Distribution in Tendon-Based Stewart Platforms. Proceedings of the ARK 02, 8th. International Symposium on Advances in Robot Kinematics,