Walking with a Double Support Phase Sylvain Miossec and Yannick Aoustin
October 14-15, Metz, 2004
Workshop on Humanoid and Legged Robots
Five link biped’s diagram Seven generalized coordinates Four actuators.
xt, yt
d2
G3
Leg 2
G2 d3 d4
Leg 1 G4
d1
a y x
S
Under actuated in Single Support: Five useful generalized coordinates
G1
R1
October 14-15, Metz, 2004
R2
Over actuated in double support Three useful generalized coordinates
Workshop on Humanoid and Legged Robots
Dynamic Model A(q ) X + H (q, q ) = DΓ Γ + D1 R1 + D2 R2 The tip of the stance legs does not move d j ( X ) = const V j = D j (q )T X = 0 ( j = 1,2) V = D (q )T X + H (q, q ) = 0 j
October 14-15, Metz, 2004
j
cj
Workshop on Humanoid and Legged Robots
Definition of a Reference Motion (1/3) Single support phase Objectif: To take into account of the under actuation δ i ,SS (α ) = ai 0 + ai1α + ai 2α 2 + ai 3α 3 + ai 4α 4 (i = 1,...,4)
These prescribed variables and the undriven variableα lead to a simple semi-inverse system no term of torques
October 14-15, Metz, 2004
σ = − Mg( xG ( α ) − xS ) σ COM of α = f (α ) the Biped
Workshop on Humanoid and Legged Robots
Definition of a Reference Motion (2/3) σ
Angular momentum σ =
Passive Impact
4
∑ k =1
f k ( δ i )δ k + f 5 ( δ i )α
+−X − ) = D1 ( q )I R + D2 ( q )I R A( q ) = ( X 1 2
If the reference trajectories are exactly tracked a new formulation of the impact: + −
α = bα
October 14-15, Metz, 2004
Workshop on Humanoid and Legged Robots
Definition of a Reference Motion (3/3) The double support phase
δ i , DS (α ) = ai 0 + ai1α + ai 2α 2 + ai 3α 3 (i = 1,2) The biped is over-actuated: it is possible to use the absolute time
α DS ( t ) = a0 + a1t + a2t 2 + a3t 3 Optimization process to obtain a cyclic reference motion with single and double support October 14-15, Metz, 2004
Workshop on Humanoid and Legged Robots
Control of α DS in Double Support Control law
α max ( α ,α ) if α ( α ) − α c ( α ) < 0 ⎧ ⎪ α = ⎨ α min ( α ,α ) if α ( α ) − α c ( α ) > 0 ⎪ ) if α ( α ) − α c ( α ) = 0 ⎩α c ( α ,α Minimal ground reaction
Constraints Maximal friction
⎧ Rïy ≥ Rïy ,min ( i = 1,2 ) ⎪ ⎨− f max Rïy ≤ Rix ≤ f max Rïy ( i = 1,2 ) ⎪ (j=1,...,4) ⎩−Γ max ≤ Γ j ≤ Γ max
Maximal torque October 14-15, Metz, 2004
Workshop on Humanoid and Legged Robots
Two simplex problems
After several manipulations
α mini ( α ,α ) = min α ⎧ α ,R2 x ⎪ ⎪ E( α ) α + F( α ,α ) + G( α )R2 x + P ≤ 0 ⎪ ⎨ ⎪ α max ( α ,α ) = max α ⎪ α ,R2 x ⎪ E( α ) α + F( α ,α ) + G( α )R2 x + P ≤ 0 ⎩
combining reference motion, dynamic problem and constraints October 14-15, Metz, 2004
Workshop on Humanoid and Legged Robots
representative results (1/2) Converging motion starting from null velocity
Cyclic motion
October 14-15, Metz, 2004
Workshop on Humanoid and Legged Robots
representative results (1/2) Convergence in one step
Poincaré map at the beginng of double support phase October 14-15, Metz, 2004
Poincaré map at the beginng of single support phase
Workshop on Humanoid and Legged Robots
An Example of Motion
Starting from a stopped position October 14-15, Metz, 2004
Workshop on Humanoid and Legged Robots
Conclusion • An Efficient control law in double support for stabilization of the walk of a biped. • Determination of the one step convergence. • It is possible to start from a stooped position. • Perspective: commutation between different cyclic motions to start, to walk or to stop... October 14-15, Metz, 2004
Workshop on Humanoid and Legged Robots
