Lecture 6: Kinematics: Velocity Kinematics - the Jacobian
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1 Lecture 6: Kinematics: Velocity Kinematics - the Jacobian Skew Symmetric Matrices c Anton Shiriaev. 5EL158: Lecture 6 p. 1/12
2 Lecture 6: Kinematics: Velocity Kinematics - the Jacobian Skew Symmetric Matrices Linear and Angular Velocities of a Moving Frame c Anton Shiriaev. 5EL158: Lecture 6 p. 1/12
3 Angular Velocity for Describing Rotation around Fixed Axis When a rigid body rotates around a fixed axis Every point of the body moves in a circle c Anton Shiriaev. 5EL158: Lecture 6 p. 2/12
4 Angular Velocity for Describing Rotation around Fixed Axis When a rigid body rotates around a fixed axis Every point of the body moves in a circle As the body rotates a perpendicular from any point of the body to the axis sweeps the same an angle θ c Anton Shiriaev. 5EL158: Lecture 6 p. 2/12
5 Angular Velocity for Describing Rotation around Fixed Axis When a rigid body rotates around a fixed axis Every point of the body moves in a circle As the body rotates a perpendicular from any point of the body to the axis sweeps the same an angle θ If k is a unit vector in the direction of the axis of rotation, then the angular velocity is given by ω = d dt θ k c Anton Shiriaev. 5EL158: Lecture 6 p. 2/12
6 Angular Velocity for Describing Rotation around Fixed Axis When a rigid body rotates around a fixed axis Every point of the body moves in a circle As the body rotates a perpendicular from any point of the body to the axis sweeps the same an angle θ If k is a unit vector in the direction of the axis of rotation, then the angular velocity is given by ω = d dt θ k The linear velocity of any point of the body is then v = ω r where r is the vector from the origin, which is assumed to lie on the axis. c Anton Shiriaev. 5EL158: Lecture 6 p. 2/12
7 Skew Symmetric Matrices An n n matrix S is said to be skew symmetric and denoted S so(n) if S + S T = 0 c Anton Shiriaev. 5EL158: Lecture 6 p. 3/12
8 Skew Symmetric Matrices An n n matrix S is said to be skew symmetric and denoted S so(n) if S + S T = 0 The matrix equation written for each item of S so(n) is s ij + s ji = 0, i = 1,..., n, j = 1,..., n c Anton Shiriaev. 5EL158: Lecture 6 p. 3/12
9 Skew Symmetric Matrices An n n matrix S is said to be skew symmetric and denoted S so(n) if S + S T = 0 The matrix equation written for each item of S so(n) is s ij + s ji = 0, i = 1,..., n, j = 1,..., n { s ii = 0 and s ij = s ji i j } c Anton Shiriaev. 5EL158: Lecture 6 p. 3/12
10 Skew Symmetric Matrices An n n matrix S is said to be skew symmetric and denoted S so(n) if S + S T = 0 The matrix equation written for each item of S so(n) is s ij + s ji = 0, i = 1,..., n, j = 1,..., n { s ii = 0 and s ij = s ji i j } For example, if n = 3, then any S so(3) has the form S = c Anton Shiriaev. 5EL158: Lecture 6 p. 3/12
11 Skew Symmetric Matrices An n n matrix S is said to be skew symmetric and denoted S so(n) if S + S T = 0 The matrix equation written for each item of S so(n) is s ij + s ji = 0, i = 1,..., n, j = 1,..., n { s ii = 0 and s ij = s ji i j } For example, if n = 3, then any S so(3) has the form S = = 0 s 3 s 2 s 3 0 s 1 s 2 s 1 0 c Anton Shiriaev. 5EL158: Lecture 6 p. 3/12
12 Skew Symmetric Matrices (Cont d) Given a = a x, a y, a z ] T, define 0 a z a y S ( a) = a z 0 a x a y a x 0 c Anton Shiriaev. 5EL158: Lecture 6 p. 4/12
13 Skew Symmetric Matrices (Cont d) Given a = a x, a y, a z ] T, define 0 a z a y S ( a) = a z 0 a x a y a x 0 Property: S (α a + β c) = αs ( a ) + βs ( c ) c Anton Shiriaev. 5EL158: Lecture 6 p. 4/12
14 Skew Symmetric Matrices (Cont d) Given a = a x, a y, a z ] T, define 0 a z a y S ( a) = a z 0 a x a y a x 0 Property: Property: S (α a + β c) = αs ( a ) + βs ( c ) S ( a) p = a p for any a, p R 3 c Anton Shiriaev. 5EL158: Lecture 6 p. 4/12
15 Skew Symmetric Matrices (Cont d) Given a = a x, a y, a z ] T, define 0 a z a y S ( a) = a z 0 a x a y a x 0 Property: Property: S (α a + β c) = αs ( a ) + βs ( c ) S ( a) p = a p for any a, p R 3 S ( a) p = 0 a z a y a z 0 a x a y a x 0 p x p y p z = p z a y p y a z p x a z p z a x p y a x p x a y c Anton Shiriaev. 5EL158: Lecture 6 p. 4/12
16 Skew Symmetric Matrices (Cont d) Property: RS ( a) R T = S ( R a ) for any R SO(3), a R 3 c Anton Shiriaev. 5EL158: Lecture 6 p. 5/12
17 Skew Symmetric Matrices (Cont d) Property: RS ( a) R T = S ( R a ) for any R SO(3), a R 3 R S ( a) R T p ] = R a R T p ] = R ] a b c Anton Shiriaev. 5EL158: Lecture 6 p. 5/12
18 Skew Symmetric Matrices (Cont d) Property: RS ( a) R T = S ( R a ) for any R SO(3), a R 3 R S ( a) R T p ] = R a R T p ] = R ] a b = R a R b c Anton Shiriaev. 5EL158: Lecture 6 p. 5/12
19 Skew Symmetric Matrices (Cont d) Property: RS ( a) R T = S ( R a ) for any R SO(3), a R 3 R S ( a) R T p ] = R a R T p ] = R ] a b = R a R b = R a R ( R T p ) = R a p c Anton Shiriaev. 5EL158: Lecture 6 p. 5/12
20 Skew Symmetric Matrices (Cont d) Property: RS ( a) R T = S ( R a ) for any R SO(3), a R 3 R S ( a) R T p ] = R a R T p ] = R ] a b = R a R b = R a R ( R T p ) = R a p = S (R a) p c Anton Shiriaev. 5EL158: Lecture 6 p. 5/12
21 Skew Symmetric Matrices (Cont d) Property: RS ( a) R T = S ( R a ) for any R SO(3), a R 3 R S ( a) R T p ] = R a R T p ] = R ] a b = R a R b = R a R ( R T p ) = R a p = S (R a) p Property: x T Sx = 0 for any S so(3), x R 3 c Anton Shiriaev. 5EL158: Lecture 6 p. 5/12
22 Derivative of Rotation Matrix Consider a function of a scalar variable θ R(θ) SO(3) c Anton Shiriaev. 5EL158: Lecture 6 p. 6/12
23 Derivative of Rotation Matrix Consider a function of a scalar variable θ R(θ) SO(3) How to compute d dθ R(θ)? c Anton Shiriaev. 5EL158: Lecture 6 p. 6/12
24 Derivative of Rotation Matrix Consider a function of a scalar variable θ R(θ) SO(3) How to compute d dθ R(θ)? For any θ the matrix R(θ) is a rotation, then R(θ)R T (θ) = R T (θ)r(θ) = I c Anton Shiriaev. 5EL158: Lecture 6 p. 6/12
25 Derivative of Rotation Matrix Consider a function of a scalar variable θ R(θ) SO(3) How to compute d dθ R(θ)? For any θ the matrix R(θ) is a rotation, then Therefore R(θ)R T (θ) = R T (θ)r(θ) = I d dt R(θ)RT (θ) ] = d dt R(θ) ] RT (θ)+r(θ) d dt RT (θ) ] = d dt I = 0 c Anton Shiriaev. 5EL158: Lecture 6 p. 6/12
26 Derivative of Rotation Matrix Consider a function of a scalar variable θ R(θ) SO(3) How to compute d dθ R(θ)? For any θ the matrix R(θ) is a rotation, then Therefore R(θ)R T (θ) = R T (θ)r(θ) = I d dt R(θ)RT (θ) ] = d ] dt R(θ) RT (θ) +R(θ) d dt }{{} = S RT (θ) ] = d dt I = 0 c Anton Shiriaev. 5EL158: Lecture 6 p. 6/12
27 Derivative of Rotation Matrix Consider a function of a scalar variable θ R(θ) SO(3) How to compute d dθ R(θ)? For any θ the matrix R(θ) is a rotation, then R(θ)R T (θ) = R T (θ)r(θ) = I Therefore d R(θ)RT (θ) ] = d ] dt dt R(θ) RT (θ) + R(θ) d RT (θ) ] = dt }{{}}{{} = S = S T d dt I = 0 c Anton Shiriaev. 5EL158: Lecture 6 p. 6/12
28 Derivative of Rotation Matrix Consider a function of a scalar variable θ R(θ) SO(3) How to compute d dθ R(θ)? For any θ the matrix R(θ) is a rotation, then R(θ)R T (θ) = R T (θ)r(θ) = I Therefore d R(θ)RT (θ) ] = d ] dt dt R(θ) RT (θ) + R(θ) d RT (θ) ] = dt }{{}}{{} = S = S T d dt I = 0 d dt R(θ) ] = SR(θ), S so(3) c Anton Shiriaev. 5EL158: Lecture 6 p. 6/12
29 Derivative of Rotation Matrix (Example 4.2) If R(θ) = R z,θ, that is the basic rotation around the axis z, then cos θ sin θ 0 cos θ sin θ 0 S = dθ d ] Rz,θ R T z,θ = d dθ sin θ cos θ 0 sin θ cos θ T c Anton Shiriaev. 5EL158: Lecture 6 p. 7/12
30 Derivative of Rotation Matrix (Example 4.2) If R(θ) = R z,θ, that is the basic rotation around the axis z, then cos θ sin θ 0 cos θ sin θ 0 S = dθ d ] Rz,θ R T z,θ = d dθ sin θ cos θ 0 sin θ cos θ = sin θ cos θ 0 cos θ sin θ 0 cos θ sin θ 0 sin θ cos θ T c Anton Shiriaev. 5EL158: Lecture 6 p. 7/12
31 Derivative of Rotation Matrix (Example 4.2) If R(θ) = R z,θ, that is the basic rotation around the axis z, then cos θ sin θ 0 cos θ sin θ 0 S = dθ d ] Rz,θ R T z,θ = d dθ sin θ cos θ 0 sin θ cos θ = = sin θ cos θ 0 cos θ sin θ 0 cos θ sin θ 0 sin θ cos θ = 0 a z a y a z 0 a x a y a x 0 = S ( a) T c Anton Shiriaev. 5EL158: Lecture 6 p. 7/12
32 Derivative of Rotation Matrix (Example 4.2) If R(θ) = R z,θ, that is the basic rotation around the axis z, then cos θ sin θ 0 cos θ sin θ 0 S = dθ d ] Rz,θ R T z,θ = d dθ sin θ cos θ 0 sin θ cos θ = = sin θ cos θ 0 cos θ sin θ 0 cos θ sin θ 0 sin θ cos θ = 0 a z a y a z 0 a x a y a x 0 = S ( a) T = S( k), k = 0, 0, 1] T c Anton Shiriaev. 5EL158: Lecture 6 p. 7/12
33 Lecture 6: Kinematics: Velocity Kinematics - the Jacobian Skew Symmetric Matrices Linear and Angular Velocities of a Moving Frame c Anton Shiriaev. 5EL158: Lecture 6 p. 8/12
34 Angular Velocity If R(t) SO(3) is time-varying and has a time-derivative, then d R(t) = S(t)R(t) = S(ω(t))R(t), dt S( ) so(3) c Anton Shiriaev. 5EL158: Lecture 6 p. 9/12
35 Angular Velocity If R(t) SO(3) is time-varying and has a time-derivative, then d R(t) = S(t)R(t) = S(ω(t))R(t), dt The vector ω(t) will be the angular velocity S( ) so(3) of rotating frame wrt to the fixed frame at time t. c Anton Shiriaev. 5EL158: Lecture 6 p. 9/12
36 Angular Velocity If R(t) SO(3) is time-varying and has a time-derivative, then d R(t) = S(t)R(t) = S(ω(t))R(t), dt The vector ω(t) will be the angular velocity S( ) so(3) of rotating frame wrt to the fixed frame at time t. Consider a point p rigidly attached to a moving frame, then p 0 (t) = R1 0 (t) p1 c Anton Shiriaev. 5EL158: Lecture 6 p. 9/12
37 Angular Velocity If R(t) SO(3) is time-varying and has a time-derivative, then d R(t) = S(t)R(t) = S(ω(t))R(t), dt The vector ω(t) will be the angular velocity S( ) so(3) of rotating frame wrt to the fixed frame at time t. Consider a point p rigidly attached to a moving frame, then p 0 (t) = R1 0 (t) p1 Differentiating this expression we obtain d dt p 0 (t) ] = d dt R 0 1 (t) ] p 1 = S(ω(t))R 0 1 (t)p1 = ω(t) R 0 1 (t)p1 = ω(t) p 0 (t) c Anton Shiriaev. 5EL158: Lecture 6 p. 9/12
38 Addition of Angular Velocities An angular velocity is a free vector { } ωi,j k (t) it corresponds to d dt Ri j (t) expressed in k-coordinate frame c Anton Shiriaev. 5EL158: Lecture 6 p. 10/12
39 Addition of Angular Velocities An angular velocity is a free vector { } ωi,j k (t) it corresponds to d dt Ri j (t) expressed in k-coordinate frame Consider two moving frames and the fixed one all with the common origin R 0 2 (t) = R0 1 (t)r1 2 (t) c Anton Shiriaev. 5EL158: Lecture 6 p. 10/12
40 Addition of Angular Velocities An angular velocity is a free vector { } ωi,j k (t) it corresponds to d dt Ri j (t) expressed in k-coordinate frame Consider two moving frames and the fixed one all with the common origin R 0 2 (t) = R0 1 (t)r1 2 (t) Taking time derivative of the left and right sides we have d dt R0 2 (t) = d dt R 0 1 (t) ] R 1 2 (t) + R0 1 (t) d dt R 1 2 (t) ] c Anton Shiriaev. 5EL158: Lecture 6 p. 10/12
41 Addition of Angular Velocities An angular velocity is a free vector { } ωi,j k (t) it corresponds to d dt Ri j (t) expressed in k-coordinate frame Consider two moving frames and the fixed one all with the common origin R 0 2 (t) = R0 1 (t)r1 2 (t) Taking time derivative of the left and right sides we have d dt R0 2 (t) = d dt R 0 1 (t) ] R 1 2 (t) + R0 1 (t) d dt = S(ω 0 0,2 (t))r0 2 R 1 2 (t) ] c Anton Shiriaev. 5EL158: Lecture 6 p. 10/12
42 Addition of Angular Velocities An angular velocity is a free vector { } ωi,j k (t) it corresponds to d dt Ri j (t) expressed in k-coordinate frame Consider two moving frames and the fixed one all with the common origin R 0 2 (t) = R0 1 (t)r1 2 (t) Taking time derivative of the left and right sides we have d dt R0 2 (t) = d dt R 0 1 (t) ] R 1 2 (t) + R0 1 (t) d dt R 1 2 (t) ] = S(ω0,2 0 (t))r0 2 ] ] = S(ω0,1 0 (t))r0 1 (t) R2 1 (t) + R0 1 S(ω (t) 1,2 1 (t))r1 2 (t) c Anton Shiriaev. 5EL158: Lecture 6 p. 10/12
43 Addition of Angular Velocities An angular velocity is a free vector { } ωi,j k (t) it corresponds to d dt Ri j (t) expressed in k-coordinate frame Consider two moving frames and the fixed one all with the common origin R 0 2 (t) = R0 1 (t)r1 2 (t) Taking time derivative of the left and right sides we have d dt R0 2 (t) = d dt R 0 1 (t) ] R 1 2 (t) + R0 1 (t) d dt = S(ω 0 0,2 (t))r0 2 R 1 2 (t) ] ] = S(ω0,1 0 (t)) R0 1 (t)r1 2 }{{ (t) +R1 0 } (t) S(ω1,2 1 (t))r1 2 (t) = R2 0 (t) c Anton Shiriaev. 5EL158: Lecture 6 p. 10/12
44 Addition of Angular Velocities An angular velocity is a free vector { } ωi,j k (t) it corresponds to d dt Ri j (t) expressed in k-coordinate frame Consider two moving frames and the fixed one all with the common origin R 0 2 (t) = R0 1 (t)r1 2 (t) Taking time derivative of the left and right sides we have d dt R0 2 (t) = d dt R 0 1 (t) ] R 1 2 (t) + R0 1 (t) d dt = S(ω 0 0,2 (t))r0 2 R 1 2 (t) ] = S(ω0,1 0 (t))r0 2 (t) + R0 1 (t)s(ω1 1,2 (t)) R0 1 R 0 (t)t 1 }{{ (t) R2 1 } (t) = I c Anton Shiriaev. 5EL158: Lecture 6 p. 10/12
45 Addition of Angular Velocities An angular velocity is a free vector { } ωi,j k (t) it corresponds to d dt Ri j (t) expressed in k-coordinate frame Consider two moving frames and the fixed one all with the common origin R 0 2 (t) = R0 1 (t)r1 2 (t) Taking time derivative of the left and right sides we have d dt R0 2 (t) = d dt R 0 1 (t) ] R 1 2 (t) + R0 1 (t) d dt = S(ω 0 0,2 (t))r0 2 R 1 2 (t) ] = S(ω0,1 0 (t))r0 2 (t) + R0 1 (t)s(ω1 1,2 (t))r0 1 R }{{ (t)t 1 0 } (t)r1 2 (t) = S(R1 0 (t)ω1 1,2 (t)) c Anton Shiriaev. 5EL158: Lecture 6 p. 10/12
46 Addition of Angular Velocities An angular velocity is a free vector { } ωi,j k (t) it corresponds to d dt Ri j (t) expressed in k-coordinate frame Consider two moving frames and the fixed one all with the common origin R 0 2 (t) = R0 1 (t)r1 2 (t) Taking time derivative of the left and right sides we have d dt R0 2 (t) = d dt R 0 1 (t) ] R 1 2 (t) + R0 1 (t) d dt = S(ω 0 0,2 (t))r0 2 R 1 2 (t) ] = S(ω 0 0,1 (t))r0 2 (t) + S(R0 1 (t)ω1 1,2 (t)) R0 1 (t)r1 2 (t) }{{} = R 0 2 (t) c Anton Shiriaev. 5EL158: Lecture 6 p. 10/12
47 Addition of Angular Velocities An angular velocity is a free vector { } ωi,j k (t) it corresponds to d dt Ri j (t) expressed in k-coordinate frame Consider two moving frames and the fixed one all with the common origin R 0 2 (t) = R0 1 (t)r1 2 (t) Taking time derivative of the left and right sides we have d dt R0 2 (t) = d dt R 0 1 (t) ] R 1 2 (t) + R0 1 (t) d dt R 1 2 (t) ] = S(ω0,2 0 (t))r0 2 ] = S(ω0,1 0 (t)) + S(R0 1 (t)ω1 1,2 (t)) R 0 2 (t) c Anton Shiriaev. 5EL158: Lecture 6 p. 10/12
48 Addition of Angular Velocities An angular velocity is a free vector { } ωi,j k (t) it corresponds to d dt Ri j (t) expressed in k-coordinate frame Consider two moving frames and the fixed one all with the common origin R 0 2 (t) = R0 1 (t)r1 2 (t) Taking time derivative of the left and right sides we have d dt R0 2 (t) = d dt R 0 1 (t) ] R 1 2 (t) + R0 1 (t) d dt R 1 2 (t) ] = S(ω0,2 0 (t))r0 2 ] = S(ω0,1 0 (t)) + S(R0 1 (t)ω1 1,2 (t)) R 0 2 (t) S(ω 0 0,2 (t)) = S(ω0 0,1 (t)) + S(R0 1 (t)ω1 1,2 (t)) c Anton Shiriaev. 5EL158: Lecture 6 p. 10/12
49 Addition of Angular Velocities (Cont d) The relation S(ω 0 0,2 (t)) = S(ω0 0,1 (t)) + S(R0 1 (t)ω1 1,2 (t)) together with the property: S(a) + S(c) = S(a + c), imply that the angular velocity can be computed as ω 0 0,2 (t) = ω0 0,1 (t) + R0 1 (t)ω1 1,2 (t) = ω0 0,1 (t) + ω0 1,2 (t) c Anton Shiriaev. 5EL158: Lecture 6 p. 11/12
50 Addition of Angular Velocities (Cont d) The relation S(ω 0 0,2 (t)) = S(ω0 0,1 (t)) + S(R0 1 (t)ω1 1,2 (t)) together with the property: S(a) + S(c) = S(a + c), imply that the angular velocity can be computed as ω 0 0,2 (t) = ω0 0,1 (t) + R0 1 (t)ω1 1,2 (t) = ω0 0,1 (t) + ω0 1,2 (t) Given n-moving frames with the same origins as for fixed one R 0 n (t) = R0 1 (t)r1 2 (t) Rn 1 n (t) d dt R0 n (t) = S(ω0 0,n (t))r0 n (t) ω 0 0,n (t) = ω0 0,1 (t) + ω0 1,2 (t) + ω0 2,3 (t) + + ω0 n 1,n (t) = ω 0 0,1 + R0 1 ω1 1,2 + R0 2 ω2 2,3 + + R0 n 1 ωn 1 n 1,n c Anton Shiriaev. 5EL158: Lecture 6 p. 11/12
51 Computing a Linear Velocity of a Point Given moving and fixed frames related by a homogeneous transform H1 0 (t) = R0 1 (t) o0 1 (t) 0 1 that is, coordinates of each point of moving frame are p 0 (t) = R1 0 (t)p1 + o 0 1 (t) c Anton Shiriaev. 5EL158: Lecture 6 p. 12/12
52 Computing a Linear Velocity of a Point Given moving and fixed frames related by a homogeneous transform H1 0 (t) = R0 1 (t) o0 1 (t) 0 1 that is, coordinates of each point of moving frame are p 0 (t) = R1 0 (t)p1 + o 0 1 (t) Hence d dt p0 (t) = d dt R 0 1 (t) ] p 1 + d dt o 0 1 (t) ] c Anton Shiriaev. 5EL158: Lecture 6 p. 12/12
53 Computing a Linear Velocity of a Point Given moving and fixed frames related by a homogeneous transform H1 0 (t) = R0 1 (t) o0 1 (t) 0 1 that is, coordinates of each point of moving frame are p 0 (t) = R1 0 (t)p1 + o 0 1 (t) Hence d dt p0 (t) = d dt R 0 1 (t) ] p 1 + d dt = S(ω 0 1 (t))r0 1 p1 + d dt = ω 0 1 (t) (R0 1 p1 ) + d dt o 0 1 (t) ] o 0 1 (t) ] o 0 1 (t) ] c Anton Shiriaev. 5EL158: Lecture 6 p. 12/12
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