Robotics. Islam S. M. Khalil. September 18, German University in Cairo

Transcription

1 Robotics German University in Cairo September 18, 2016

2 Vector Functions A vector v in a reference frame A depends on a scalar verifiable q. We can say that v is a vector function of q in A. Otherwise, v is independent of q in A. Figure: P is dependent on q 1 and q 2 in S and independent on q 3 in S.

3 Vector Functions A vector v may be a function of a variable q in one reference frame, but be independent of q in another reference frame. Figure: A gyroscope.

4 Vector Functions P is the position vector from point O to point P of C. P is a function of q 1 both in A and B, but is independent of q 1 in C. Figure: A gyroscope.

5 Vector Functions P is a function of q 2 both A, but is independent of q 2 in B and in C. Figure: A gyroscope.

6 Vector Functions P is independent of q 3 both A, B, and C, but is a function of q 3 in D. Figure: A gyroscope.

7 Vector Functions In a reference frame A, a vector function v of n scalar variables q 1,..., q n, let a 1, a 2, and a 3 be a set of nonparallel, noncoplanar unit vectors fixed in A. Then there exist three unique scalar functions v 1, v 2, and v 3 of q 1,..., q n such that v = v 1 a 1 + v 2 a 2 + v 3 a 3 (1) When a 1, a 2, and a 3 are mutually perpendicular unit vectors, then it follows from (1) that the v i is given by Therefore, (1) can be rewritten as follow: v i = v a i (2) v = v a 1 a 1 + v a 2 a 2 + v a 3 a 3 (3)

8 Vector Functions a 1, a 2, a 3 and b 1, b 2, b 3 are mutualy perpendicular unit vectors fixed in A and B, respectively. Vector p can be expressed both as and as p = α 1 a 1 + α 2 a 2 + α 3 a 3 (4) p = β 1 b 1 + β 2 b 2 + β 3 b 3 (5) Figure: A gyroscope. α i and β i for i = 1, 2, 3 are functions of q 1, q 2, and q 3.

9 Vector Functions To determine the functions α i and β i, note that, if C has a radius R, one can proceed from O to P by moving through the distances R cos q 1 and R sin q 1 in the directions of b 2 and b 3, respectively, therefore p = R cos q 1 b 2 + R sin q 1 b 3 (6) Comparing (5) to (6), we find that β 1 = 0 β 2 = R cos q 1 β 3 = R sin q 1 (7)

10 Vector Functions Based on (4), one can write α 1 = p a 1 = R (cos q 1 b 2 a 1 + sin q 1 b 3 a 1 ) (8) α 2 = p a 2 = R (cos q 1 b 2 a 2 + sin q 1 b 3 a 2 ) (9) α 3 = p a 3 = R (cos q 1 b 2 a 3 + sin q 1 b 3 a 3 ) (10) We also know that b 2 a 1 = 0 b 2 a 2 = 1 b 2 a 3 = 0 (11) b 3 a 1 = cos q 2 b 3 a 2 = 0 b 3 a 3 = sin q 2 (12) Therefore, the functions α i of p are α 1 = R sin q 1 cos q 2 α 2 = R cos q 1 α 3 = R sin q 1 sin q 2 (13)

11 First Derivatives If v is a vector function of n scalar variables q 1,..., q n in a reference frame A, then n vectors called first partial derivatives of v in A and denoted by the symbols A v or A (v), (r = 1,..., n) q r q r are defined as follows: Let a 1, a 2, and a 3 be any any nonparallel, noncoplanar unit vectors fixed in A, and let v i be the a i measure number of v then A v q r 3 i=1 v i q r a i, (r = 1,..., n) (14) When v is regarded as a vector function of only a single scalar variable in Afor instance the time tthen this definition reduces to that of the ordinary derivative of v with respect to t in A, to 3 A dv dq r dv i dq r a i, (r = 1,..., n) (15) i=1

12 First Derivatives The vector p possesses partial derivatives with respect to q 1, q 2, and q 3 in each of the reference frames A, B, and C. To form A p q r, (r = 1,..., n) we use (14) as follows: A p [ ] [ ] = R sin q 1 cos q 2 a 1 + R cos q 1 a 2 + q r q r q r [ ] R sin q 1 sin q 2 a 3 (16) q r Consequently, A p q 1 = R (cos q 1 cos q 2 a 1 sin q 1 a 2 + cos q 1 sin q 2 a 3 ) (17) A p q 2 = R sin q 1 ( sin q 2 a 1 + cos q 2 a 3 ) (18) A p q 3 = 0 (19) p is independent of q 3 in A.

13 First Derivatives A p q 1 = R (cos q 1 cos q 2 a 1 sin q 1 a 2 + cos q 1 sin q 2 a 3 ) (20) A p q 2 = R sin q 1 ( sin q 2 a 1 + cos q 2 a 3 ) (21) A p q 3 = 0 (22) p is independent of q 3 in A.

14 First Derivatives Proceeding similarly to determine B p q r, (r = 1,..., n), we obtain B p q 1 = R ( sin q 1 b 2 + cos q 1 b 3 ) p is independent of q 2 and q 3 in B. B p q 2 = 0 B p q 3 = 0 (23) Figure: A gyroscope.

15 First Derivatives Similarly to determine C p q r, (r = 1,..., n), we obtain p is independent of q 1, q 2 and q 3 in C. C p q r = 0 (24) Figure: A gyroscope.

16 First Derivatives Suppose now that q i are specified as explicit functions of time t The α i can be expressed as q 1 = t q 2 = 2t q 3 = 3t α 1 = R sin t cos 2t α 2 = R cos t α 3 = R sin t sin 2t and the ordinary derivative of p with respect to t in A is seen to be given by A dp dt = dα 1 dt a 1 + dα 2 dt a 2 + dα 3 dt a 3 (25) = R[(cos t cos 2t 2 sin t sin 2t)a 1 (sin t)a 2 + (cos t sin 2t + 2 sin t cos 2t)a 3 ]

17 First Derivatives while the ordinary derivative of p with respect to t in B is B dp dt = R ( sin tb 2 + cos tb 3 ) (26) Finally, C dp dt = 0 (27) Figure: A gyroscope.

18 Thank You Thank You! Questions please