Manipulator Dynamics Case Study

As shown in Fig. A.2, the position vector on a continuous curve can be expressed as r(s) where s is the curve length parameter. When the curve is smooth enough, we can differentiate r(s) two times with s to obtain equation (A.17), where T is a unit vector pointing to the tangential direction of the curve and N is a unit vector which is perpendicular to T . ( 0)in equation (A.17) is called the curve curvature. Note that by this definition, N always points to the inner side of the curve as showing in

Fig. A.2.

8<

: dr=ds = T dr2=ds2 = dT =ds = N

(A.17)

Fig. A.2. The position r(s), unit tangential T and unit normalN vectors on a curve.

Furthermore, we can define a new unit vector B by using equation (A.18). By these definitions, equation (A.19)

It means the Runge-Kutta method provides a universal numerical solution for all

The main results summarized here are only for a quick reference.

The motion of a mechanical system can be formulated by Euler-Lagrange equations expressed in equation (A.26), where qi expresses the generalized coordinate variable which may be a joint angle or a displacement. The integer n is the joint number of the generalized coordinates required to describe the system and i is the generalized force acted on joint i which may be force or torque. The L called as the

Lagrangian, is the difference of the kinetic and potential energy of the system. The

Euler-Lagrange equations are equivalent to the formulations derived by using Newton’s

Second Law. Lagrangian approach is more preferred in analyzing the motion of a complex system such as manipulators. d dt

@L

@q_i

@L

@qi

= i; i = 0; 1; 2;


; n (A.26)

To apply Euler-Lagrange equations, the kinetic and potential energy of the system need to be expressed by using qi and q_i where i = 1; 2;


; n. Also, the initial value of qi(0) and q_i(0) must be assigned.

In general, the kinetic K and the potential energy P of a manipulator can