Humanoid robot walking with variable center-of-mass height The variable-height inverted pendulum is a point mass model that extends the linear inverted pendulum model by removing its constant center-of-mass height assumption. This extension introduces a new control input, the leg stiffness, that enables a "height-variation strategy" for balance recovery and locomotion …
In this note, we consider the motion of a rigid body B \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol{H}} \def\bfI{\boldsymbol{I}} \def\bfJ …
The wheeled inverted pendulum is a nonlinear inverted pendulum attached to a wheel rolling without slipping on the floor. In this post, we review the assumptions that define it, derive its equation of motion ℓ θ ¨ = g sin ( θ ) − r ¨ cos ( θ ) , \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B …
When dealing with a planar surface contact, such as the foot of a humanoid on a flat ground or the palm of its hand against a wall, friction fights against the two surfaces rotating agaisnt each other, that is, rotation along the contact normal. This rotation is often called twisting …
In inverse kinematics, our control problem is to bring task residuals , also known as task errors , to zero. If we denote by e ( q ) \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG …
One topic that comes out often ( e.g. 1 , 2 , 3 , 4 ) in technical discussions around linear inverted pendulum tracking is the regulation of contact forces by damping control. Let us review the working assumptions and model behind this choice. Flexibility model ¶ Robots from the HRP series have inherited from …
In this post, we are taking a look at the kinematics of a simple leg with two links and one knee. Both links have the same length ℓ \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF …
There are two ways model predictive control (MPC) has been applied to legged locomotion so far: open loop and closed loop MPC. In both cases, a model predictive control (numerical optimization) problem is derived from a model of the system and solved, providing a sequence of actions that can be …
Consider the planar legged model depicted to the right. The robot's mass is concentrated at its center of mass G \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH …
Revolute joints are the most common ones in legged robots. Their equations of motion involve both joint coordinates q \def\bfA{\boldsymbol{A}} \def\bfB{\boldsymbol{B}} \def\bfC{\boldsymbol{C}} \def\bfD{\boldsymbol{D}} \def\bfE{\boldsymbol{E}} \def\bfF{\boldsymbol{F}} \def\bfG{\boldsymbol{G}} \def\bfH{\boldsymbol …