Adaptive Control of A Spring-Mass Hopper İsmail Uyanık*, Uluç Saranlı § and Ömer Morgül* *Department of Electrical and Electronics Engineering § Department of Computer Engineering Bilkent Dexterous Robotics and Locomotion Group (BDRL)

2 Introduction ANIMAL MODEL PHYSICAL MODEL MATHEMATICAL MODEL

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 The SLIP Model The SLIP model consists of a point mass attached to a massless leg with a linear spring k and viscous damping d. - System alternates between stance and flight phases. - Apex: The highest point in the flight phase. We define nth apex state as, - We collect relevant dynamic parameters of the system in a single vector as Control parameters: -The leg angle, - Touchdown leg length, - Liftoff leg length,

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 Measurement and estimation of dynamic system parameters such as k&d is a challenging problem. Spring and damping constants have a possibly time-varying and unpredictable nature. Most existing methods assume the perfect knowledge of dynamic parameters, and ignore the effects of miscalibrated parameters. Motivated by the work in this area, we present a new model-based adaptive control method for running with SLIP model, emphasizing on-line estimation of unknown or miscalibrated system parameters. Fortunately, this issue is not confined to the control of legged locomotion and received considerable attention from the adaptive control community. The SLIP Model

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 The SLIP Model We define a discrete apex return map which depends on the dynamic parameters p of the system as, Problem: The stance dynamics of SLIP under the effect of gravity is nonintegrable. Solution: Approximate analytical stance maps. We use analytical approximations proposed by Ankarali (AAS approximations), which can successfully incorporate the effects of both damping and gravity and define the approximate return map as, Consequently, we capture dependence of apex velocity and height to p which these return map depends on.

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 Adaptive Control of SLIP Running Existing Method - Given a desired apex state X*, inversion of the apex return map for height and lateral velocity components of the state yields the controller, Problem: The approximate return map and hence its inversion can only rely on possibly inaccurate parameter estimates for spring and damping constants. As shown, the estimates are generally obtained through calibration experiments but may not provide sufficiently good accuracy.

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 Adaptive Control of SLIP Running Our adaptive control algorithm relies on once-per-step corrections to these estimates based on the difference between predicted and measured apex states for each apex state. Here, g is the available approximate return map that can predict the apex state outcome of a single step given the previous apex state and associated control input. We consider two alternatives for approximate predictor model g, Exact SLIP Model (ESM): This alternative uses g = f, computed through numerical simulation of SLIP dynamics. The associated Jacobian J is also computed numerically. Approximate Analytic Solution (AAS): This option uses Ankarali’s approximate analytic solution as a predictor for SLIP trajectories. The associated Jacobian J AAS is analytically derived through straight-forward differentiation.

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 Adaptive Control of SLIP Running ESM option is useful for accurate identification of the dynamic parameters of the system. AAS option is useful in eliminating steady-state tracking errors for the gait-level control of SLIP running. Regardless of the predictor, an apex state prediction error is computed as The goal of our adaptive control approach is to bring the steady-state value of prediction error to zero, It also yields steady-state parameter estimates as The Jacobian matrices relate infinitesimal changes in the apex state predictions to infinitesimal changes in the dynamic system parameters with We propose the parameter update strategy where Ke < 1 is a gain coefficient to tune convergence and prevent oscillatory behavior.

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 Simulation apex goal and parameter ranges individual single step runs “Ground truth”: Both flight and stance dynamics of the physical SLIP plant were simulated in Matlab using a fourth-order, adaptive time-step Runge-Kutta integrator. Simulation were run until steady-state was reached with a tolerance of 10^-4 in the norm of apex state. Simulation Studies

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 Performance Criteria Percentage apex tracking and parameter estimation errors System identification performance Tracking performance Notable tracking performance improvement with AAS predictor Notable system identification with ESM predictor Performance Analysis

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 Accurate Control with the AAS Predictor Example SLIP Simulation Starts with a non-adaptive controller (dark shaded region ) 20% error in both the spring and damping constants Substantial steady-state error Adaptive controller with the AAS predictor was started around t = 2s Steady-state error is quickly eliminated A step change in the apex goal was given around t = 4:55s Steady-state tracking remain accurate

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 Results Accurate Control with the AAS Predictor Adaptive controller ensures g p (Xn,u)  fp(Xn,u) by controlling p Deadbeat controller ensures f p (Xn,u)  X* by controlling u Eventually, g p (Xn,u)  X* The dependence of average errors and their std on the initial deviations of the spring and damping constants. AAS predictor works best for accurate control in all cases. Both AAS and ESM predictors outperform non-adaptive controller

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 Dynamic tracking performance - Sinusoidal paths for desired height and velocities - Outperforms non-adaptive controller Accurate Control with the AAS Predictor Proposed controller can maintain accurate tracking even for dynamic goal settings and not just for a single static target.

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 System Identification with the ESM Predictor Example SLIP Simulation Starts with a non-adaptive controller (dark shaded region ) 20% error in both the spring and damping constants Substantial estimation error Adaptive controller with the ESM predictor was started around t = 2s Accurate estimation of dynamic system parameters A step change in the apex goal was given around t = 4:55s Estimation performance remain accurate

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 System Identification with the ESM Predictor Results The dependence of average estimation errors and their std on the initial values of the spring and damping constants ESM allows more accurate estimation of unknown system parameters at the expense of steady-state tracking. AAS predictor also performs well yielding errors well below 10-15% errors expected from manual calibration alone. Eventually,

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 Conclusion & Future Works A novel adaptive control algorithm to both support on-line identification of unknown dynamic system parameters and improve steady-state tracking performance of previously proposed control algorithms for the well known SLIP model. We showed that ESM method based on numerical integration of SLIP dynamics is capable of accurate system identification, whereas ASM method based on the analytical approximations allows elimination of steady-state tracking errors for a deadbeat controller based on the same approximations. Our long term goal is to design legged platforms that can reactively negotiate rough terrain. The applicability of this purpose critically depends on our ability to accurately estimate associated parameters in the mathematical models. In a near future, we will implement the method used in this paper on a monopedal platform and our future goal includes extensions of this method to more complex legged models and locomotion controllers. We showed that in all cases, our adaptive methods perform much better than the non- adaptive approach both for gait control and system identification.

Bilkent Dexterous Robotics and Locomotion Group (BDRL) 3 Acknowledgments We thank Mustafa Mert Ankaralı and Melih Çakmakçı for their ideas and support. This work was supported by National Scientific and Technological Council of Turkey (TUBITAK), through project 109E032 and İsmail Uyanık’s scholarship.

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