Robotics. Islam S. M. Khalil. September 19, German University in Cairo

Transcription

1 Robotics German University in Cairo September 19, 2016

2 Angular Velocity Let b l, b 2, and b 3 form a right-handed set of mutually perpendicular unit vectors fixed in a rigid body B moving in a reference frame A. The angular velocity of B in A, denoted by A ω B, is defined as A ω B b 1 A db 2 b 3 + b 2 A db 3 b 1 + b 3 A db 1 b 2 (1) One task facilitated by the use of angular velocity vectors is the time-differentiation of vectors fixed in a rigid body, for it enables one to obtain the first time-derivative of such a vector by performing a cross-multiplication. Specifically, if β is any vector fixed in B, then A dβ A ω B β (2)

3 Angular Velocity A ω B b 1 A db 2 A db 3 db 1 b 3 + b 2 b 1 + b 3 b 2 (3) A dβ A ω B β (4) Figure: Angular velocity of B with respect to A.

4 Angular Velocity Derivation Using dots to denote time-differentiation in A, one can rewrite (1) A ω B b 1 ḃ 2 b 3 + b 2 ḃ 3 b 1 + b 3 ḃ 1 b 2 (5) and cross-multiplication of (5) with b 1 gives We also know that A ω B b 1 b 2 b 1 ḃ 3 b 1 + b 3 b 1 ḃ 1 b 2 (6) b 2 b 3 b 1 b 3 b 1 b 2 (7) Substitution of (7) in (6) yields A ω B b 1 b 3 ḃ 3 b 1 + b 2 ḃ 1 b 2 (8)

5 Angular Velocity Derivation Moreover, time-differentiation of the equations b 1 b 1 1 and b 3 b 1 0 yields rewrite (8) ḃ 1 b 1 ḃ 3 b 1 ḃ 1 b 3 (9) A ω B b 1 b 1 ḃ 1 b 1 + b 2 ḃ 1 b 2 + b 3 ḃ 1 b 3 (10) But the right-hand member of this equation is simply a way of writing ḃ 1. A ω B b 1 ḃ1 (11) Similarly, A ω B b 2 ḃ 2 A ω B b 3 ḃ 3 (12) and, after expressing any vector β fixed in B as β β 1 b 1 + β 2 b 2 + β 3 b 3 (13)

6 Angular Velocity Derivation where β 1, β 2, and β 3 are constants, so that β β 1 ḃ 1 + β 2 ḃ 2 + β 3 ḃ 3 (14) Therefore, β β A 1 ω B b 1 + β A 2 ω B b 2 + β A 3 ω B b 3 (15) Finally we obtain β A ω B (β 1 b 1 + β 2 b 2 + β 3 b 3 ) A ω B β (16) Auxiliary Reference Frames β A ω B β (17)

7 Angular Velocity Example. Here B represents a door supported by hinges in a room A. Mutually perpendicular unit vectors a 1, a 2, and a 3 are fixed in A, with a 3 parallel to the axis of the hinges, and mutually perpendicular unit vectors b 1, b 2, and b 3 are fixed in B, with b 3 a 3.

8 Angular Velocity If θ is the radian measure of the angle between a 1 and b 1 as shown below, then a 1, a 2, and a 3 and b 1, b 2, and b 3 are related to each other as indicated in the table below b 1 b 2 b 3 a 1 cos θ -sin θ 0 a 2 sin θ cos θ 0 a The angular velocity of B in A, A ω B is given by A ω B θb 3 (18) The second time-derivative in A of the position vector from the point O to the point P, that is, of the vector β given by A β t β L 1 b 1 L 3 b 3 (19) The derivative of β in A is given by θb 3 ( L 1 b 1 L 3 b 3 ) L 1 θb 2 (20)

9 Angular Velocity A 2 β t 2 L 1 θb 2 L 1 θ A b 2 t Since b 2 is a vector fixed in B, its time-derivative in A can be found as follows: (21) A b 2 t A ω B b 2 θb 1 (22) Therefore, A 2 β ( t 2 L θ2 1 b 1 θb ) 2 (23) In more complex situations, that is, when the motion of B in A is more complicated than that of a door B in a room A, the use of the angular velocity vector as an operator which, through cross-multiplication, produces time-derivatives, is all the more advantageous.

10 Differentiation in Two Reference Frames If A and B are any two reference frames, the first time-derivatives of any vector v in A and in B are related to each other as follows: A dv B dv + A ω B v (24) where A ω B is the angular velocity of B in A. Derivation. Let us define a vector v in B so that 3 v v i b i (25) i1 The time-derivative of v in A is A dv 3 dv i b i + i1 B dv + 3 i1 v i A db i (26) 3 v A i ω B b i (27) i1

11 Differentiation in Two Reference Frames A dv 3 i1 B dv B dv B dv dv i b i i1 v i A db i (28) 3 v A i ω B b i (29) i1 + A ω B Differentiation in Two Reference Frames A dv B dv 3 v i b i (30) i1 + A ω B v (31) + A ω B v (32)

12 Auxiliary Reference Frames The angular velocity of a rigid body B in a reference frame A can be expressed in the following form involving n auxiliary reference frames A 1,..., A n : Auxiliary Reference Frames A ω B A ω A 1 + A 1 ω A A n 1 ω An + An ω B (33) The reference frames A 1,..., A n may or may not correspond to actual rigid bodies. Frequently, such reference frames are introduced as aids in analysis, but have no physical counterparts. Derivation. For any vector β fixed in B A dβ A ω B β A 1 dβ A 1 ω B β (34) A dβ A 1 dβ + A ω A 1 β (35)

13 Auxiliary Reference Frames Therefore, A ω B β A 1 ω B β + A ω A 1 β (36) Since this equation is satisfied by every β fixed in B, it implies that A ω B A 1 ω B + A ω A 1 (37) Proceeding similarly, one can verify that Auxiliary Reference Frames A ω B A ω A 1 + A 1 ω A A n 1 ω An + An ω B (38)

14 Angular Acceleration The angular acceleration A α B of a rigid body B in a reference frame A is defined as the first time-derivative in A of the angular velocity of B in A A α B A d A ω B (39) We also know that A dv B dv + A ω B v (40) A α B A d A ω B B d A ω B + A ω B A ω B B d A ω B (41) The angular velocity of B in A can always be expressed as A ω B ωk ω, where k ω is a unit vector parallel to A ω B ; similarly A α B can always be expressed as A α B αk α, where k α is a unit vector parallel to A α B. In general, k ω differs from k α.

15 Velocity and Acceleration Let p denote the position vector from any point O fixed in a reference frame A to a point P moving in A. The velocity of P in A and the acceleration of P in A, denoted by A v P and A a P 9 respectively, are defined as and A v P A dp A a P A d A v P (42) (43)

16 Velocity and Acceleration Example P 1 and P 2 designate two points connected by a line of length L and free to move in a plane B that is rotating at a constant rate ω about a line Y fixed both in B and in a reference frame A. The velocities A v P 1 and A v P 2 of P 1 and P 2 in A are to be expressed in terms of the quantities q 1, q 2, q 3 their time-derivatives q 1, q 2, q 3, and the mutually perpendicular unit vectors e x, e y, e z. Figure: Velocity.

17 Velocity and Acceleration Example The point O is fixed in A, and the position vector p 1 from O from P 1 is p 1 q 1 b x + q 2 b y (44) A v P 1 where A dp 1 B dp 1 + A ω B p 1 A dp 1 q 1 b x + q 2 b y A ω B p 1 ωb y (q 1 b x + q 2 b y ) ωq 1 b z Therefore, A v P 1 q 1 b x + q 2 b y ωq 1 b z (45) Figure: Velocity.

18 Velocity and Acceleration Example The unit vectors b x,y,z are related to the unit vectors e x,y,z e x e y e z b through x cos q 3 -sin q 3 0 b y sin q 3 cos q 3 0 b z A v P 1 ( q 1 cos q 3 + q 2 sin q 3 )e x +( q 1 sin q 3 + q 2 cos q 3 )e y ωq 1 e z which is the desired expression for the velocity of P 1 in A. Proceeding similary, one can determine the velocity of P 2 in A. Figure: Velocity.

19 Thank You Thank You! Questions please