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## Model Predictive Control for Humanoid Balance and Locomotion

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**Model Predictive Control for Humanoid Balance and Locomotion**Benjamin Stephens Robotics Institute**Compliant Balance and Push Recovery**• Full body compliant control • Robustness to large disturbances • Perform useful tasks in human environments**Motivation**• Improve the performance and usefulness of complex robots, simplifying controller design by focusing on simpler models that capture important features of the desired behavior • Enabling dynamic robots to interact safely with people in everyday uncertain environments • Modeling human balance sensing, planning and motor control to help people with disabilities**Outline**• Optimal Control Formulation • Humanoid Robot Control • Examples and Problems**Outline**• Optimal Control Formulation Formulate balance and foot placement control as an optimal control problem**Linear Inverted Pendulum Model**Assumptions: • Zero vertical acceleration • No torque about COM Constraints: • COP within the baseof support REFERENCE: Kajita, S.; Tani, K., "Study of dynamic biped locomotion on rugged terrain-derivation and application of the linear inverted pendulum mode," ICRA 1991**Optimal Control of Walking**Objective Function • Must provide footstep locations and timings • Double support is largely ignored Wieber, P.-B., "Trajectory Free Linear Model Predictive Control for Stable Walking in the Presence of Strong Perturbations," Humanoid Robots 2006**Optimal Control with Foot Placement**Next 3 Footsteps: Time of step is encoded in U0 and U1 Diedam, H., et. al., "Online walking gait generation with adaptive foot positioning through Linear Model Predictive control," IROS 2008**Optimal Step Recovery**3. 1. 2. Objective Function • Must provide footstep timing • Must decide which foot to step with • Constraints in double support are nonlinear due to variable foot location**Optimal Step Recovery**a=1e-6 b=0.1 c=0.01 d=1e-6 X0 = [0,0,0.4,-0.1] T=0.05 Tstep=0.4 N=20**Initial double support phase**Tdsp = 0.0s Tstep = 0.45s Tdsp = 0.1s Tstep = 0.35s**Outline**• Optimal Control Formulation • Humanoid Robot Control • Examples and Problems**Outline**• Humanoid Robot Control Use MPC inside feedback loopto generate desired contactforces and joint torques**Simple Biped Dynamics**Center of pressure (COP) Angular momentum Foot locations Instantaneous 3D biped dynamics form a linear system in contact forces. Center of mass (COM)**Simple Biped Inverse Dynamics**• The contact forces can be solved for generally using constrained quadratic programming Least squares problem(quadratic programming) Linear Inequality Constraints • COP under each foot • Friction**Controlling a Complex Robot with a Simple Model**• Full body balance is achieved by controlling the COM using the policyfrom the simple model. • The inverse dynamics chooses from the set of valid contact forces the forcesthat result in the desired COM motion.**General Humanoid Robot Control**Dynamics Contact constraints Control Objectives Desired COM Motion Pose Bias**Feed-forward Force Inverse Dynamics**• Pre-compute contact forces using simple model and substitute into the dynamics**Other Tasks**• Posture Control • Angular Momentum Regulation • Swing Foot Control • Task Control (e.g. lifting heavy object) Benjamin Stephens, Christopher Atkeson, "Push Recovery by Stepping for Humanoid Robots with Force Controlled Joints,"Accepted to 2010 International Conference on Humanoid Robots, Nashville, TN. Benjamin Stephens, Christopher Atkeson, "Dynamic Balance Force Control for Compliant Humanoid Robots,“ 2010 International Conference on Intelligent Robots and Systems (IROS), Taipei, Taiwan.**Outline**• Optimal Control Formulation • Humanoid Robot Control • Examples and Problems**Extensions**• Different Models • Swing Leg • Torso • Angular Momentum • Different Objective Functions • Capture Point • Minimum Variance Control • Step Time Optimization**Open Problems**• Learning from experience • Using human motion capture • Higher-level planning • State Estimation and Localization