Analytical Dynamics of Open Kinematic Chains

Analytical Dynamics of Open Kinematic Chains - Chapter 6...

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Chapter 6 Analytical Dynamics of Open Kinematic Chains 6.1 Generalized Coordinates and Constraints Consider a system of N particles: { S } = { P 1 , P 2 ,... P i ... P N } . The position vector of the i th particle in the Cartesian reference frame is r i = r i ( x i , y i , z i ) and can be expressed as r i = x i ı + y i j + z i k , i = 1 , 2 ,..., N . The system of N particles requires n = 3 N physical coordinates to specify its po- sition. To analyze the motion of the system in many cases, it is more convenient to use a set of variables different from the physical coordinates. Let us consider a set of variables q 1 , q 2 q 3 N related to the physical coordinates by x 1 = x 1 ( q 1 , q 2 q 3 N ) , y 1 = y 1 ( q 1 , q 2 q 3 N ) , z 1 = z 1 ( q 1 , q 2 q 3 N ) , . . . x 3 N = x 3 N ( q 1 , q 2 q 3 N ) , y 3 N = y 3 N ( q 1 , q 2 q 3 N ) , z 3 N = z 3 N ( q 1 , q 2 q 3 N ) . The generalized coordinates , q 1 , q 2 q 3 N , are the set of variables that can completely describe the position of the dynamical system. The confguration space is the space extended across the generalized coordinates. If the system of N particles has m constraint equations acting on it, the system can be represented uniquely by p independent generalized coordinates q k ,( k =1 ,2,. .. , p ), where p = 3 N m = n m . The number p is called the number of degrees of freedom of the system. 209
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210 6 Analytical Dynamics of Open Kinematic Chains The number of degrees of freedom is the minimum number of independent coor- dinates necessary to describe the dynamical system uniquely. The generalized veloc- ities , denoted by ˙ q k ( t ) ( k =1,2,. ., n ), represent the rate of change of the generalized coordinates with respect to time. The state space is the 2 n -dimensional space spanned by the generalized coordi- nates and generalized velocities. The constraints are generally dominant as a result of contact between bodies, and they limit the motion of the bodies upon which they act. A constraint equation and a constraint force are related with a constraint. The constraint force is the joint reaction force and the constraint equation represents the kinematics of the contact. Consider a smooth surface of equation f ( x , y , z , t )= 0 , (6.1) where f has continuous second derivatives in all its variables. A particle P is sub- jected to a constraint of moving on the smooth surface described by Eq. 6.1. The constraint equation f (x, y, z, t) = 0 represents a conFguration constraint . The motion of the particle over the surface can be viewed as the motion of an oth- erwise free particle subjected to the constraint of moving on that particular surface. Hence, f (x, y, z, t) = 0 represents a constraint equation. For a dynamical system with n generalized coordinates, a con±guration con- straint can be described as f ( q 1 , q 2 ,..., q n , t 0 . (6.2) The differential of the constraint f , given by Eq. 6.1, in terms of physical coordinates is df = f x dx + f y dy + f z dz + f t dt = 0 . (6.3) The differential of the constraint f , given by Eq. 6.2, in terms of the generalized coordinates is = f q 1 dq 1 + f q 2 2 + ... + f q n n + f t = 0 . (6.4) Equations 6.3 and 6.4 are called constraint relations in PfafFan form . A constraint in Pfaf±an form is a constraint that is represented in the form of differentials.
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Analytical Dynamics of Open Kinematic Chains - Chapter 6...

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