University of Pisa Project work for Robotics Prof. Antonio Bicchi Students: Sergio Manca Paolo Viccione WALKING ROBOT

Basic characteristic of all biped locomotion systems: – Contact of the foot with the ground surface – Gait repeatability – Interchangeability of the single- and double-support phases

WALKING ROBOT During the walk two different situations arise in sequence: – Statically stable double-support phase: both the feet in contact with the ground – Single-support phase: only one foot is in contact with the ground, small RoS The locomotion mechanism changes its structure during the single walking cycle from an open to a closed kinematic chain.

WALKING ROBOT

System Geometry (DH) Direct Dynamic: ✔ Mathematica script (Lagrange formulation) ✔ SimMechanics ✔ MSC Adams Inverse Dynamic (Newton-Eulero): ✔ Robotics Toolbox ✔ Mathematica script ✔ C++ Trajectory planning and management phases of system ✔ StateFlow Matlab ✔ C++

GEOMETRY Denavit-Hartenberg formulation Mathematica script

DIRECT DYNAMIC 1 st approach: Mathematical modeling Symbolic equations of motion with Lagrange formulation: Mathematica script:

DIRECT DYNAMIC 1 st approach: Mathematical modeling Issues – The reference frame of the base link is supposed to be inertial – Complex systems →either huge matrices or high computational complexity Tests: – Systems up to 6 DOF – System 8 DOF or more

DIRECT DYNAMIC 2 nd approach: Physical modeling SimMechanics PROS: Handle complex systems Rapid modeling by blocks CONS: Solver issues simulating contacts

DIRECT DYNAMIC 3 rd approach: MSC Adams Mechanical model Robust stiff solver Matlab control interface Rendering

INVERSE DYNAMIC Newton-Eulero Robotics toolbox does not implement: – Closed kinematic chains – Non-inertial base link – Application of external forces besides the end-effector – Breaking the backward iteration in the middle in order to compute dynamic compensation Iterative numeric algorithm suitable even for real time implementation No need for the closed form of the dynamic equations

INVERSE DYNAMIC Newton-Eulero Script Mathematica: Forward propagation of velocities and accelerations Backward propagation of forces and torques C++ implementation Speed-up over 1000 Slow even if built and distributed

TRAJECTORY PLANNING Stateflow Matlab – phase management complex – rigid C++ – simple – fast

CONTROL The system having the coordinates q as output has maximum relative degree; Full feedback linearization, by numerical inversion of the dynamic – Full dynamic decoupling – Yields an easy design of the controller for asintotic stability and tracking of trajectories.

CONTROL CONTROLLER INVERSE DYNAMIC Newton-Eulero DIRECT DYNAMIC Adams

CONTROL: the LMI's approach Pole placement in a convex region: Chilali,Gahinet '96 With

CONTROL: the LMI's approach H ∞ optimal control on frequency weighted tracking error and control action. Bound the amplitude of the control. Kalman-Yakubovich-Popov (KYP) Lemma:

CONTROLLER: Comparison Static controllerLMI's controller

CoP and ZMP Two concepts are mostly used to measure and ensure dynamic stability: – ZMP(Zero Moment Point) – CoP(Center of Pressure) Having the CoM projection within the contact area does not ensure stability for a non-static structure.

ZMP The Zero Moment Point concept is an extension of the CoM projection taking into account the inertia forces as well as the gravity forces. The ZMP is the point on the ground where the tipping moment acting on the system equals zero.

ZMP For a non-static structure: The ZMP is different w.r.t. the projection of the Center of Mass (CoM) on the ground. Having the ZMP within the contact area ensure dynamic stability.

ZMP ZMP formal definition: The ZMP is the point on the ground where the resultant moment acting on the biped, due to gravity and inertia forces, is normal to the supporting plane of the tow feet.

CoP CoP formal definition: The CoP is the point on the ground where the moment due to the resultant force of the contact forces (pressure+frictional forces) is normal to the supporting plane of the two feet.

CoP and ZMP The Newton-Eulero equations of the biped global motion can be written as: yields: – the concepts of CoP and ZMP are strictly related; – It is more convenient to measure the contact forces rather than the link's accelerations.

UNILATERAL CONSTRAINTS The actuation torques generated by NE to track the desired CoP are not always compatible with the unilateral constraint at the base. 3-Phases NE: Forward: accelerations; Backward: Base reaction; Re-forward: Compute the compatible reaction and propagate. If a correction has been made, the end effector's requested interaction in the backward phase is not going to be met.

CoP placement How much torque the ankle needs to do in order to place the CoP under the foot?

CoP placement τ CoP Backward Direct torque control: NE Backward Unilateral constraint matching NE Re-forward residual propagation up to the trunk Re-forward

Appendix A (CoP placement)

THE END Questions?....