Robot Leg Equations

#Robotics #Modeling The main equations governing the motion of the leg are:

• The Newton Equation: \(m (\ddot{c} + g) = \sum_i f_i\) where $m$ is the total mass of the robot, $c$ is the position of its center of mass (CoM), and $f_i$ are the forces acting on the leg.
• The Euler Equation: \(\dot{L} = \sum_i (p_i - c) \times f_i\) where $L$ is the angular momentum of the whole robot with respect to its CoM and is defined as: \(L = \sum_k (x_k - c) \times m_k \dot{x}_k + I_k \omega_k\) There are two phases in moving the leg:
• Flight Phase: When a legged robot is not in contact with its environment, not experiencing any contact forces, $f_i$, the Newton equation simplifies to \(\ddot{c} = - g\) and the Euler equation simplifies to \(\dot{L} = 0\) There is absolutely no possibility to control the CoM to move in any different way. However, the robot is still able to generate and control both joint motions.
• Stance Phase: In case the forces applied by the environment on the robot are due to contacts with a flat ground, one can show that the center of pressure (CoP) is bound to lie in the convex hull of the contact points

Contact Models

There are several ways to model how a legged robot contacts the ground.

• Coulomb Friction: When a contact point, $p_i$, is sliding on its contact surface, the corresponding tangential contact force, $f_i^{x,y}$ , is proportional to the normal force, $f_i^z$, in a direction opposite to the sliding motion: \(f_i^{x,y} = -\mu_0 f_i^z \frac{\dot{p}_i^{x,y}}{\parallel \dot{p}_i^{x,y} \parallel} \ ,\ if\ \dot{p}_i^{x,y} \neq 0\) with $\mu_0$ as the friction constant. When the contact point is sticking and not sliding, the norm of the tangential force is simply bounded, with the same friction coefficient: \(\parallel f_i^{x,y} \parallel \leq \mu_0 f_i^z \ ,\ if\ \dot{p}_i^{x,y} \neq 0\) This is typically referred to as the friction cone.
• Compliant Contact Models: Compliant contact models take into account the visco-elastic properties of the materials in contact in the direction orthogonal to the contact surfaces: \(f_i^z = - K_i p_i^z - \Lambda_i \dot{p}_i^z \ ,\ if\ \dot{p}_i^z \neq 0\) where $K_i$ $and $\Lambda_i$ are stiffness and damping coefficients, respectively. When there is no contact, there is no contact force: \(f_i^z = 0 \ ,\ if\ \dot{p}_i^z > 0\)
• Rigid Contact Models: Rigid contact models are simpler to introduce. Either there is contact and the normal force can take any non-negative value \(f_i^z \geq 0 \ ,\ if\ \dot{p}_i^z = 0\) or there is no contact and no contact force: \(f-i^z = 0 \ ,\ if\ \dot{p}_i^z > 0\) The rigid model can be summarized in the following way: \(f_i^z \geq 0 \, \ p_i^z \geq 0 \, \ f_i^z p_i^z = 0\) This is called a complementarity condition. With rigid contacts, the dynamics of legged robots appears to switch, depending on the contact situations. Discontinuities of the state of the robot also occur at impacts. A classical way to combine these different aspects is with a hybrid dynamical system. This approach has however some limitations, the most obvious one being its incapacity to handle properly Zeno behaviors, infinite accumulations of impacts in finite time. Impacts with multiple contacts and Zeno behaviors are not the only difficulties with rigid contact models: there is also the Painleve paradox, tangential impacts, impacts without collisions, etc. By conservatively approximating the friction cone as a polyhedron, forward dynamics can be cast as a linear complementarity problem (LCP).

To read more, you can continue to Stability of Legged Robots.

Sources:

1. Modeling and Control of Legged Robots