Presentazione standard di PowerPointProf.Fanny Ficuciello Robotics for Bi oengineering • Dynamics Lagrangian is a function of the generalized coordinates: T and U: total kinetic

Prof. Fanny Ficuciello Robotics for Bioengineering • 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)

Dynamics


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)

Lagrange formulation


kinetic energy

potential energy

lagrangian

motion equation

Example


contributions relative to the motion of each link and of each joint actuator

inertia matrix

symmetric positive definite configuration-dependent

Kinetic energy and potential energy


Lagrange equations


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

Dynamic model in the joint space


is skew-symmetric Christoffel symbols of the first type

principle of conservation of energy (Hamilton)

Properties


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

Linearity in the dynamic parameters


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


Newton–Euler formulation


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 determine

useful in simulation

inverse dynamicknown determine

useful for planning and control



Lagrange

knowing compute

and then integrating with step compute

Newton-Euler computational more efficient method

Direct kinematics


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

Operational space dynamic model


direct dynamics known determine

direct joint dynamics direct kinematics

inverse dynamicknown 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



suppose the manipulator still and not in contact with the environment

ellipsoid in the operational space

Dynamic manipulability ellipsoid


non redundant manipulator

Dynamic manipulability ellipsoid

Documents

Presentazione standard di PowerPointProf.Fanny Ficuciello Robotics for Bi oengineering • Dynamics Lagrangian is a function of the generalized coordinates: T and U: total kinetic