Towards Simulating a Mid-size Stewart Platform on a Large Hexapod Simulator

Transcription

1 AIAA Modeling and Simulation Technologies Conference - August 9, Chicago, Illinois AIAA Towards Simulating a Mid-size Stewart Platform on a Large Hexapod Simulator F.M. Nieuwenhuizen, Max Planck Institute for Biological Cybernetics, Tübingen, Germany M.M. van Paassen, M. Mulder, Delft University of Technology, Delft, The Netherlands K. Beykirch, and H.H. Bülthoff Max Planck Institute for Biological Cybernetics, Tübingen, Germany For a recent project on the influence of motion system characteristic on human perception and control behaviour, a dynamic model of the MPI Stewart platform was developed. The model parameters were estimated from measurements involving motion along a circular trajectory and frequency sweeps. Simulation results showed that the model response with optimised parameters was very close to the measured platform response. However, additional measurements are required to identify the platform mass and vertical centre of gravity position correctly. Validation of the dynamic model with platform measurements in heave showed favourable results. The dynamic model of the MPI Stewart platform will be validated further in multiple degrees-of-freedom and will be used in active closed-loop experiments. I. Introduction The Max Planck Institute for Biological Cybernetics (MPI) operates a mid-size hexapod motion platform: the MPI Stewart platform. This type of system is commonly used for flight simulation and is used in most advanced simulators for pilot training. At the MPI, the motion platform has a custom-built cabin with a visualisation system and a design that allows for modular adjustments of the projection screen and input devices. The platform is used for passive experiments involving integration of visual and vestibular cues for the perception of self-motion, and for active experiments related to behaviour of humans in closed-loop control tasks. A recent project had the goal of determining the characteristics of the MPI Stewart platform and building a dynamic model of the platform. With this dynamic model, the MPI Stewart platform can be simulated on a large research simulator with hydraulic actuators, the SIMONA Research Simulator(SRS) at Delft University of Technology. In such a setup, the motion characteristics of the MPI Stewart platform can be altered systematically. By modelling the multi-modal human perception and control behaviour in active closed-loop control tasks,, the influence of the motion characteristics can be assessed. Previously, the dynamic characteristics of the MPI Stewart platform were determined. A uniform and systematic approach based on the AGARD report was taken to arrive at results that should allow for direct comparison with similar measurements on other motion platforms. The measurement on the MPI Stewart platform revealed a fixed time delay of ms, that was attributed to the software setup, and a rather restricted operating range based on the default platform filters that were implemented Ph.D. candidate, Max Planck Institute for Biological Cybernetics, P.O. Box 69, 7 Tübingen, Germany; frank.nieuwenhuizen@tuebingen.mpg.de. Student member AIAA. Associate Professor, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 558, 6GB, Delft, The Netherlands; m.m.vanpaassen@tudelft.nl. Member AIAA. Professor, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 558, 6GB, Delft, The Netherlands; m.mulder@tudelft.nl. Member AIAA. Research Scientist, Max Planck Institute for Biological Cybernetics, P.O. Box 69, 7 Tübingen, Germany; karl.beykirch@tuebingen.mpg.de. Professor and Director, Max Planck Institute for Biological Cybernetics, P.O. Box 69, 7 Tübingen, Germany; heinrich.buelthoff@tuebingen.mpg.de. Member AIAA. of Copyright 9 by Max Planck Institute for Biological Cybernetics. American Published Institute by of the Aeronautics American Institute and of Astronautics Aeronautics and Astronautics, Inc., with permission.

2 by the platform manufacturer. These findings resulted in modifications to the platform software and filters, allowing for an increase in operating range and performance. Based on measurements on the MPI Stewart platform, a dynamic model was determined. The model will be validated with measurements performed on MPI Stewart platform and the SRS. After that, several experiments with active closed-loop control tasks will be performed, first in a single degree-of-freedom, second by including an additional degree of freedom. By systematically manipulating the characteristics of the motion cues, insight can be gained in motion characteristics that are important for simulator motion fidelity. In the current paper, the model of the MPI Stewart platform is presented. First, the MPI Stewart platform is introduced and measurements with the current platform software and filters are introduced. Second, the dynamic model for the motion platform and measurements for the determination of its parameters are presented. Finally, the results of the parameter model are given and conclusions are drawn. A. MPI Stewart platform II. Platform performance measurements The MPI Stewart platform is a mid-size hexapod motion system (Maxcue 65, Motionbase, United Kingdom). The parallel actuators are electric and support payloads up to kg. See Fig. for an impression and an overview of the specifications. The MPI Stewart platform is equipped with a custom-built cabin that allows for modular adjustments of input devices. A flat or curved screen with a field of view of approximately 7 horizontally and 5 vertically can be used as visual display. The platform is controlled through a light-weight in-house software framework that handles the network communication between various computers. Z X Y Feature Specification Payload kg Actuator stroke 5 mm Actuator length resolution.6 μm Surge range 9 mm Sway range 86 mm Heave range 5 mm Pitch range +/ deg Roll range ±8 deg Yaw range ± deg Figure : The MPI Stewart platform. B. Platform describing function The manufacturer of the motion system of the MPI Stewart platform has implemented a default low-pass platform filter for each degree-of-freedom (DoF). The standard break frequency of the filters was set at Hz but was increased to Hz for the current measurements. The high-bandwidth filters are described by the following equation, which includes a time delay τ: H platform = ( + π s) e τs =.5s +.8s+ e τs. () To check if the platform response was similar to the high-bandwidth platform filters, the motion response of the platform was measured by performing by a describing function measurement. The measurement was performed with two multi-sine signals containing five and six frequency/amplitude pairs, respectively, similar to previous measurements. The platform response was calculated from IMU measurements and gives the relation between the input and output in the driven DoF. The describing function measured for the heave DoF is given in Fig.. The figure shows that, when taking time delay τ equal to 5 ms, the measured describing function closely matches the analytical describing function given in Eq. (). The small deviations in the measured amplitude are associated with measurement noise. In previous measurements a time delay of ms was found. However, an improved software setup was used in the current measurements, which explains the decreased time delay. of

3 , db Hplatform analytical measured -6 - Hplatform, deg ω, rad/s Figure : Platform describing function in heave. With the implementation of the Hz platform filters, the bandwidth of the MPI Stewart platform was extended considerably. Also, the time delay in the motion system was reduced significantly by introducing a new software setup. III. Modelling the MPI Stewart platform With the insight gained from the performance measurements and the significant extension of the bandwidth of the motion system, a dynamic model of the MPI Stewart platform can be constructed. A. Inverse and forward kinematics The kinematics of the Stewart platform concern the relation between the platform pose (and derivatives thereof) and the actuator lengths, velocities, and accelerations. 5 7 The platform pose and its derivatives are defined as follows: 5 x = x y z φ θ ψ, x = [ c ω ] = x y z p q r, x = [ c ω ] = x y z p q r. () Here, the translational DoF in surge is given as x, in sway is y, and in heave is z. The platform positions are grouped in vector c, which is the location of the Upper Gimbal Point (UGP). The platform roll, pitch, and yaw angle are denoted as φ, θ, and ψ, respectively, and have associated angular velocities of the cabin given as p, q, and r. The angular velocities are grouped in vector ω. With the inverse kinematics, actuator length, velocity, and acceleration can be calculated from the platform pose and its derivatives. As the Stewart platform is a parallel motion system, the inverse kinematics can be calculated analytically. For the inverse position kinematics, the following equation holds: 5,6 l = c+t a c b, () where l contains the vectors between the leg attachment points on the base and cabin frame, T is the rotation matrix between the base and cabin frame, and where a and b are the location matrices of the gimbals of the cabin and base, respectively. The values for the latter two variables are specified by the platform manufacturer and given in Table. The cabin reference frame, denoted by superscript c, goes through the UGP, which is centre of the upper frame of the motion system. By differentiating Eq. (), the inverse rate kinematics can be found. These can be written as follows: of

4 Table : Gimbal locations of the MPI Stewart platform. base cabin leg x (m) y (m) x (m) y (m) l = J lx x, () where J lx is the platform Jacobian matrix. The columns of J lx specify the velocities required from the actuators to get unit velocity of the platform and its value can be calculated analytically. 6 In a specific system configuration and pose, the Jacobian is a measure for the kinematic efficiency of the platform motion. It can also be used to derive platform performance measures, such as dexterity, manipulability, and stiffness. 6 The reverse process to the inverse kinematics is to determine the platform pose from actuator length measurements and is called the forward kinematics. For a general Stewart platform, multiple solutions can be found for the forward kinematics problem. 7 Thus, the forward kinematics can not be solved analytically and a numerical iterative technique must be applied. In general, a Newton-Raphson method is used to solve the forward kinematic problem. It is formulated as: x i+ = x i +(J lx ( x i )) ( lmeas l( x i ) ). (5) The initial guess x should be sufficiently close to the actual platform pose and could, for example, be the desired platform pose. The iterative process should be repeated until a solution is found with an acceptable tolerance between the measured and calculated actuator lengths. In practical applications, a tolerance level of 6 m is reached in - iterations. B. Dynamic platform model The dynamics of the Stewart platform describe the relation between the generalised force/torque vector and the generalised position, velocity and acceleration. The inverse dynamics are used to calculate actuator forces from position and its derivatives. For this, an analytic solution exists, similar to the inverse kinematics. The forward dynamics are used to calculate the motion of the Stewart platform given the actuator forces. When assuming the platform cabin as a rigid body, and disregarding the inertial forces of the actuators, the Stewart platform dynamics can be modelled as follows: [ L n TA c L n ] [ f a = m c I T TIc ct ][ c ω ] + [ T ΩTIc ct ][ c ω ] [ m c g ]. (6) Here, L n is a matrix that contains the normalised actuator vectors, A c is a matrix that holds the platform gimbal positions in the platform reference frame, fa are the actuator forces, m c is the cabin mass, I is the identity matrix, I c c is the platform inertia tensor in the cabin reference frame, Ω is a skew-symmetric matrix that contains the platform angular rates, and g is the gravity vector. A reduced form of the model is given as: J T lx f a = M c ( x) x+c c ( x, x) x+g c, (7) where the influence of the mass matrix M c, the coriolis and centripetal effects C c, and the gravity G c are clearly separated. The Jacobian J lx is used to transform the actuator forces into the platform coordinate frame. of

5 C. Platform model block diagram The dynamic model of a Stewart platform given in Eq. (7) has parameters: the platform mass m c, the position of the centre of gravity x cog, y cog, z cog, and the values of the inertia tensor I c c. As the cabin is symmetric in the forward-backward vertical plane, we can assume that the cross products of inertia I xy = I yx and I yz = I zy equal zero. Furthermore, I xz = I zx is assumed to be small with respect to the principal moments I xx, I yy, and I zz and therefore negligible. This means that the cabin reference frame is considered as the principle axis of the cabin. q ref s e e K p K d F legs Geometry I c c Platform dynamics output q q x K s m c CoG F static J T lx mc g Figure : The platform model block diagram. A block diagram of the complete platform model is given in Fig.. Based on the motion system documentation, the controller of the platform has the form of a PD-controller and thus regulates leg lengths and leg velocities. The proportional gain is given as K p and the differential gain as K d. The relative contribution of the controller gains is known, but not the exact values. Therefore, K p and K d are expressed in terms of a general controller gain K c by the following expressions: K p = K c and K d = 7 K c. (8) The controller gain K c is unknown and is estimated in the platform parameter optimisation. This leaves a total of eight parameters to be determined, which are given in Table with boundary conditions. The PD-controller of the platform is only responsible for the dynamic platform motion. As can be seen in Fig., a feedback of static forces is implemented based on the current platform position. The gain K s on the feedback of static forces is to account for the bias in leg length found in measurements. It is calculated based on the leg following errors, the mass in the model and the controller gain K c and generally has a value between.995 and.999. Table : Platform model parameters and boundary conditions. Parameter Min. Max. m c, kg 5 6 x cog, m -.. y cog, m -.. z cog, m -.5. I xx, kg m 5 I yy, kg m 5 I zz, kg m 5 K c, N/m 5 5 D. Model assumptions To arrive at the platform model equations and block diagram described in the previous sections, several assumptions were made. The first assumption, that the products of inertia are zero, has already been described. Furthermore, the gimbal point locations are based on specifications from the manufacturer. On the MPI Stewart platform, small deviations of the gimbal locations are likely and have an impact on 5 of

6 the calculated pose and Jacobian matrix. Also the platform leg length measurements are assumed to be properly calibrated by the manufacturer. Furthermore, the platform cabin is taken as a rigid body and the mass and inertia properties of the actuators are not taken into account. A large relative contribution of the actuator mass to the total mass could warrant inclusion of these effects in the platform model. Finally, hysteresis in the actuators is not modelled. E. Measurement for platform mass and inertia properties All platform motion starts from the neutral point in the middle of the operational workspace. In this point, the actuators are in the middle of their stroke. Thus, it is important to determine the platform characteristics around this point. Therefore, the describing function measurement used sinusoidal input signals around the neutral point of the motion system. To gain insight into the motion platform response over a continuous frequency range of interest, frequency sweeps in the acceleration domain were used. By using an exponentially increasing sweep frequency, it was ensured that more time is spent at lower frequencies, where the characteristic periods are longer, than at higher frequencies. 8 The frequency sweeps are implemented as follows: u sweep = Asin(θ(t)) with θ(t) = Ts ω(t)dt, (9) where A is the sweep amplitude, and T s is the sweep time. The frequency progression ω(t) between the minimum frequency ω min and the maximum frequency ω max is given by: ω(t) = ω min +C [exp(c t/t s ) ](ω max ω min ). () The values C =. and C =.87 have been found to be suitable for a wide range of applications. 8 Furthermore, the starting frequency was held constant for a period of time that corresponds to the inverse of the starting frequency and then progressed according to the exponential function. The frequency sweep signal was faded in and out to ensure that the initial and final conditions for the velocity and acceleration signals in each measurement were zero. The final frequency progression, the acceleration signal, and the corresponding velocity and position signals for a general frequency sweep with amplitude 6 deg/s is shown in Fig.. ω(t), rad/s asweep, deg/s vsweep, deg/s xsweep, deg Figure : A general motion platform frequency sweep. From Fig. it is clear that there is a small drift from the neutral position during the frequency sweep. This is caused by the fade-in on the acceleration frequency sweep. However, this drift progresses slowly and is not regarded to have an adverse effect on the measurements. As the motion system is non-linear and platform properties might depend on the platform pose, it is also important to investigate the system behaviour away from the neutral position. Therefore, measurements were performed with a circular movement in the horizontal plane. A circle radius of.5 m was chosen, with a go-around time of s. 6 of

7 t = s t = 67 s t = 5 s leg length, m.5.. x, m t = 5 s t =,9 s t = 8 s t = s t = 55 s y, m following error, m (a) The trajectory in the horizontal plane. Figure 5: Circular platform motion. (b) The length and following error of leg. Note the differences in y-axis scaling. During the platform motion, leg lengths and leg following errors, which are the differences between the commanded, analytical leg lengths and the measured leg lengths, were obtained. In Fig. 5b these measurements during a circular platform movement are given for leg. From the figure it is clear that there was a static following error. This was found in all platform legs and is attributed to the mass of the platform cabin. The figure also clearly shows the influence of the platform controller, with peaks in the following error at the extremities of the motion profile. Furthermore, the jumps in the following error measurements signify the presence of hysteresis in the legs, which is not accounted for in the dynamic model. F. Determination of model parameters The platform model parameters, given in Table with minimum and maximum values considered in the optimisation, were determined by minimising the squared difference between the measured leg lengths and the leg lengths calculated with the platform model. As the platform model is non-linear, an optimisation procedure might find a local minimum instead of a global minimum. Therefore, a grid search was performed to find the optimum parameter vector. From inspection of Eq. (6), there is a clear separation between the platform model parameters for translational and rotational DoF s. In the translational degrees of freedom, the moments of inertia play a minor role, and thus these can be determined separately. Therefore, first the mass m c, position of the centre of gravity and the controller gain K c were determined from the circle motion measurements. After this, the moment of inertia of each axis was determined independently with frequency sweeps in each rotational DoF. IV. Results In this section, the results from the model parameter optimisation will be discussed. Also, results from the initial validation of the MPI Stewart platform model are presented. A. Platform model parameters The platform mass m c, controller gain K c, and the horizontal position of the centre of gravity, x cog and y cog, were determined from the circle motion measurement. The vertical position of the centre of gravity z cog does not have much influence in this type of motion. Therefore, z cog is determined from frequency sweep measurements in all rotational DoF s, together with the mass moments of inertia. The final results of the platform parameter optimisation are summarised in Table. The results from the grid search optimisation showed that for increasing platform mass m c and 7 of

8 Table : The estimated platform model parameters. Parameter m c x cog y cog z cog I xx I yy I zz K c K s kg m m m kg m kg m kg m N/m - Value increasing controller gain K c multiple local minima were found that were not numerically different from the global minimum. Therefore, the correct m c -K c pair needed to be determined from measurements of physical properties of the MPI Stewart platform. The cabin mass m c was estimated as approximately 5 kg, given the construction of the cabin. The controller gain K c that corresponded to this platform mass equals 5 N/m. To validate the platform mass estimate, the cabin will be detached from the motion system in the future to determine its mass. Thefeedbackgain K s associatedwiththevaluesform c andk c equals.997. Thehorizontal position of the centre of gravity was found.5 m in front of the UGP, which can be attributed to the display screen, and slightly away from the vertical symmetry plane by -.5 m. x, m (a) Surge. y, m (b) Sway. z, m commanded measured simulated (c) Heave. φ, deg (d) Roll. θ, deg (e) Pitch. ψ, deg (f) Yaw. Figure 6: Platform commanded, measured and simulated motion for a circle motion. Note the differences in y-axis scaling, the axis for heave spans μm. Simulations with the estimated parameters are shown in Fig. 6. This figure also includes the commanded platform motion and the platform motion reconstructed from the measured actuator lengths. The simulated platform position is accurate on a sub-millimetre level. From the heave position it is clear that the feedback gain K s allowed the platform model to track the static bias in the measurements, but that the controller needed some time to settle. The measured values for the rotational DoF s were very small, but the platform model captured the trends in the measurements. The vertical position of the centre of gravity z cog was estimated from frequency sweeps in the roll and pitch degree of freedom. However, no clear influence on the measurements was found. Additionally, the construction of the platform cabin is such that most mass is concentrated around the UGP as the construction that encloses the cabin is made from aluminium and cloth. This minimises the contribution of the vertical position of the centre of gravity. Therefore, a value of.5 m was assumed based on physical properties of the platform cabin, but validation with additional measurements is required. The results of a frequency sweep in yaw is given in Fig. 7. The simulated response in the driven axis is very close to the measured response and the moment of inertia for the z-axis I zz was estimated as 5 kg m. From the figure it is clear that the simulated response in the undriven axes showed similar trends as were found in the measurements. This indicates that the estimate for the horizontal position of the centre of gravity was correct. The moments of inertia for the x-axis and y-axis, I xx and I yy, were estimated from frequency sweeps 8 of

9 x, m (a) Surge. y, m (b) Sway. z, m commanded measured simulated (c) Heave. φ, deg (d) Roll. θ, deg (e) Pitch. ψ, deg (f) Yaw. Figure 7: Platform commanded, measured and simulated motion for a frequency sweep in yaw. Note the differences in y-axis scaling, the axis for heave spans μm. in roll and pitch. For both axes a similar result of 85 kg m was found. The model response in the driven axis for both frequency sweeps was very similar to the measured platform response. However, the model response in undriven axes was not simulated correctly and generally too high. Also, based on evaluation of the physical construction of the MPI Stewart platform with most mass close to the UGP, it was concluded that the values for I xx and I yy are too high. Additional measurements are required to investigate the optimisation results for I xx and I yy. In general, the dynamic model of the MPI Stewart platform with optimised parameters showed responses quite similar to the measurement data that was used for estimating the parameters. In the next section, the model will be validated with data from independent measurements. B. Platform model validation In order to be valid, a model must predict measurement data that was not used in the model determination process. In this case, the dynamic model of the MPI Stewart platform was used to simulate measurements with sinusoidal inputs in heave. The results are given in Fig. 8 for a particular time range in one of the measurements. It is clear that the dynamic model followed the measurement data from the MPI Stewart platform very well. The response of the driven axis was very precise and the response in undriven axes showed similar behaviour as the measurement data. The hysteresis in the platform actuators is clearly noticeable in the platform yaw response (see Fig. 8f). The model predicted no yaw motion, and there was actually very little in the platform response apart from moments in time where the actuators changed direction. At these moments there is a step in the model response. It can be concluded that the dynamic model of the MPI Stewart platform gives good simulation results with the independent measurement data. However, further validation in other DoF s is required, and extending the measurements to multiple DoF s is considered. V. Conclusion A dynamic model of the MPI Stewart platform was developed and its parameters were estimated using motion along a circular trajectory and using frequency sweeps. Simulations with the platform model showed close resemblance to measurement data. By determining the platform response function, a time delay of 5 ms was found. It was found that the model parameters for platform mass and controller gain were dependent on each other, the model response did not change when increasing both parameters. This can be solved by 9 of

10 x, m (a) Surge. y, m (b) Sway. z, m commanded measured simulated (c) Heave. φ, deg (d) Roll. θ, deg (e) Pitch. ψ, deg (f) Yaw. Figure 8: Platform commanded, measured and simulated motion for a validation measurement in heave. Note the differences in y-axis scaling. fixating the parameter for platform mass after weighing the cabin of the MPI Stewart platform. The position of the centre of gravity was found close to the centre of the upper platform frame. This was expected as most the cabin mass is centred around this location. However, the vertical position of the centre of gravity did not have a large influence in the measurements and therefore a unique solution could not be found. Additional measurements will be performed to get a better estimate for this parameter. The dynamic model will be validated further with additional measurements on the MPI Stewart platform and also on the SIMONA Research Simulator. After validation, the dynamic model will be used in active closed-loop experiments to determine the influence of the motion system characteristics on human perception and control behaviour. References Nieuwenhuizen, F. M., Zaal, P. M. T., Mulder, M., van Paassen, M. M., and Mulder, J. A., Modeling Human Multichannel Perception and Control Using Linear Time-Invariant Models, Journal of Guidance, Control, and Dynamics, Vol., No., July Aug. 8, pp Zaal, P. M. T., Pool, D. M., Chu, Q. P., van Paassen, M. M., Mulder, M., and Mulder, J. A., Modeling Human Multimodal Perception and Control Using Genetic Maximum Likelihood Estimation, Journal of Guidance, Control, and Dynamics, Vol., No., July Aug. 9, pp Nieuwenhuizen, F. M., Beykirch, K. A., Mulder, M., van Paassen, M. M., Bonten, J. L. G., and Bülthoff, H. H., Performance Measurements on the MPI Stewart Platform, Proceedings of the AIAA Modeling and Simulation Technologies Conference and Exhibit, Honolulu (HI), No. AIAA8-65, 8 Aug. 8. Lean, D. and Gerlach, O. H., AGARD Advisory Report No. : Dynamics Characteristics of Flight Simulator Motion Systems, Tech. Rep. AGARD-AR, North Atlantic Treaty Organization, Advisory Group for Aerospace Research and Development, Koekebakker, S. H., Model Based Control of a Flight Simulator Motion System, Doctoral dissertation, Faculty of Aerospace Engineering, Delft University of Technology,. 6 Advani, S. K., Nahon, M. A., Haeck, N., and Albronda, J., Optimization of Six-Degrees-of-Freedom Motion Systems for Flight Simulators, Journal of Aircraft, Vol. 6, No. 5, Sept. Oct. 999, pp Harib, K. and Srinivasan, K., Kinematic and dynamic analysis of Stewart platform-based machine tool structures, Robotica, Vol.,, pp Tischler, M. B. and Remple, R. K., Aircraft and Rotorcraft System Identification, American Institute of Aeronautics and Astronautics, Inc., 8 Alexander Bell Drive, Reston, VA, 6. of