Dynamics describe the relationship between the joint actuator torques and the motion of the structure important role for simulation of motion (test control strategies) analysis of manipulator structures (mechanical design of prototype arms) design of control algorithms Lagrange formulation, systematic formulation independently of the reference coordinate frame Newton Euler formulation, computationally more efficient since it exploits the typically open structure of the manipulator kinematic chain (yields the model in a recursive form)
Lagrange formulation Lagrangian is a function of the generalized coordinates: T and U: total kinetic energy and potential energy of the system generalized coordinates describing the configuration of the manipulator can be chosen as Lagrange equations Iis the generalized force associated to the generalized coordinate (non conservative forces)
Example kinetic energy potential energy lagrangian motion equation
Kinetic energy and potential energy contributions relative to the motion of each link and of each joint actuator inertia matrix symmetric positive definite configuration-dependent
Lagrange equations
Dynamic model in the joint space the coefficient represents the moment of inertia at Joint i axis, in the current manipulator configuration, when the other joints are blocked the coefficient accounts for the effect of acceleration of Joint j on Joint j the term represents the centrifugal effect induced on Joint i by velocity of Joint j the term represents the Coriolis effect induced on Joint i by velocities of Joints j and k
Properties is skew-symmetric Christoffel symbols of the first type principle of conservation of energy (Hamilton)
Linearity in the dynamic parameters mass of the link and of the motor first inertia moment of the augmented link Inertia tensor of the augmented link moment of inertia of the rotor
Newton Euler Formulation is based on a balance of all the forces acting on the generic link of the manipulator this leads to a set of equations whose structure allows a recursive type of solution a forward recursion is performed for propagating link velocities and accelerations followed by a backward recursion for propagating forces
Newton Euler formulation
Direct dynamics and inverse dynamics the Lagrange formulation has the following advantages: it is systematic and of immediate comprehension it provides the equations of motion in a compact analytical form containing the inertia matrix, the matrix in the centrifugal and Coriolis forces, and the vector of gravitational forces such a form is advantageous for control design it is effective if it is wished to include more complex mechanical effects such as flexible link deformation the Newton Euler formulation has the following advantage: it is an inherently recursive method that is computationally efficient
Direct dynamics and inverse dynamics direct dynamics known useful in simulation inverse dynamic known determine determine useful for planning and control
Direct kinematics Lagrange knowing compute and then integrating with step compute Newton-Euler computational more efficient method
Operational space dynamic model describes the relationship between the generalized forces acting on the manipulator and the number of minimal variables chosen to describe the end-effector position and orientation in the operational space Lagrange formulation using operational space variables allows a complete description of the system motion only in the case of a nonredundant manipulator, when the above variables constitute a set of generalized coordinates in terms of which the kinetic energy, the potential energy, and the nonconservative forces doing work on them start from the joint space model equivalent end-effector forces γ
Operational space dynamic model second order differential equation transformations motion equation
Direct dynamics and inverse dynamics direct dynamics known direct joint dynamics direct kinematics inverse dynamic known determine determine solution (kinematic redundancy) inverse kinematics joint space inverse dymanics solution (dynamic redundancy) dynamic model in the operational space formal solution that allows redundancy resolution at dynamic level
Dynamic manipulability ellipsoid suppose the manipulator still and not in contact with the environment ellipsoid in the operational space
Dynamic manipulability ellipsoid non redundant manipulator