Rigid body dynamics. Basilio Bona. DAUIN - Politecnico di Torino. October 2013

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1 Rigid body dynamics Basilio Bona DAUIN - Politecnico di Torino October 2013 Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

2 Multiple point-mass bodies Each mass is described by position r i = [ x i y i z i ] T and velocity v i = [ v xi v yi v zi ] T, represented in a reference frame Rb. Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

3 Consider a pure rotation around a point O, for simplicity the origin of the reference frame R b. If the angular velocity of the system is ω, every mass will acquire a linear velocity v i v i (t) = ω(t) r i (t) = S(ω(t))r i (t) The linear momentum p i (t) is defined as p i (t) = m i v i (t) In the following slides the symbol p indicates the linear momentum an not the cartesian pose or the point position. Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

4 Moment of a Force Given a point mass located in a point P represented by r and a force f applied to it, the moment of the force with respect to the point O is given by r f = r m dv dp = r dt dt Applying the derivative rules we obtain r f = r dp dt = d dr (r p) dt dt p = d dt (r p) v p }{{} 0 where the last term is always zero, since p and the v are always collinear. The vector h = r p is called angular momentum. Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

5 Since r f is often called torque produced by the force with respect to O, we can write Note the analogy τ = d dt h f = d dt p = ṗ τ = d dt h = ḣ We define generalized momentum H the vector that includes both momenta and generalized force the following [ f F = = τ] d [ [ṗ ] p = = dt h] ḣ Ḣ Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

6 Angular momentum and inertia matrix Given a point O in space and a point mass m i with position r i, the angular momentum h i, is h i (t) = r i (t) p i (t) = r i (t) (m i v i (t)) Replacing v i = ω r i and omitting for simplicity the time dependence, we obtain h i = m i (r i (ω r i )) Summing up all the N point masses contributions, we have the total angular momentum h = N i=1 h i = i m i (r i (ω r i )) Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

7 Recalling the triple cross product property, we can write h = ( ) m i (r T i r i )ω (r T i ω)r i i Replacing the square norm r i 2 with the symbol ri 2 and writing explicitly r T i ω = (x i ω x +y i ω y +z i ω z ) we obtain h = i m i r 2 i ω x (x i ω x +y i ω y +z i ω z ) x i ω y ω z y i z i or h = h x h y = i h z m i (ri 2 xi 2)ω x m i x i y i ω y m i x i z i ω z m i x i y i ω x +m i (ri 2 yi 2)ω y m i y i z i ω z m i x i z i ω x m i y i z i ω y +m i (ri 2 zi 2)ω z Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

8 The relation can be written in matrix form as h = Γ xx,i Γ xy,i Γ xz,i Γ yx,i Γ yy,i Γ yz,i ω x ω y = i Γ zx,i Γ zy,i Γ zz,i ω z i The new matrix Γ is defined as Γ = Γ xx Γ xy Γ xz Γ yx Γ yy Γ yz = Γ zx Γ zy Γ zz i where and Γ i ω Γ xx,i Γ xy,i Γ xz,i Γ yx,i Γ yy,i Γ yz,i = Γ zx,i Γ zy,i Γ zz,i i Γ xx,i = m i (ri 2 xi 2 ) = m i (yi 2 +zi 2 ) = m i dx 2 Γ yy,i = m i (ri 2 yi 2 ) = m i (xi 2 +zi 2 ) = m i dy 2 Γ zz,i = m i (ri 2 zi 2 ) = m i(xi 2 +yi 2 ) = m idz 2 Γ i Γ xy,i = Γ yx,i = m i x i y i Γ xz,i = Γ zx,i = m i x i z i Γ yz,i = Γ zy,i = m i y i z i Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

9 Therefore h = Γ xx Γ xy Γ xz Γ yx Γ yy Γ yz ω x ω y = Γω Γ zx Γ zy Γ zz ω z or better, restating the time dependence h(t) = Γ(t)ω(t) The matrix Γ is called the inertia matrix or inertia tensor and has on the main diagonal the inertia moments defined as Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

10 Another possible representation of h and Γ can be obtained using the properties of the skew-symmetric matrices: h = i m i (r i (ω r i )) = i m i S(r i )S(ω)r i Recalling that S(ω)r i = S(r i )ω, one obtains: h = i m i S(r i )S(ω)r i = i m i S(r i )S(r i )ω = i m i S 2 (r i )ω ( ) h = i m i S 2 (r i ) ω = Γω with Γ = i m i S 2 (r i ) Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

11 Distributed Mass Systems inertia matrix as: Γ = obtaining the inertia moments and the inertia products V ρ(x,y,z) y2 +z 2 xy xz xy x 2 +z 2 yz dv xz yz x 2 +y 2 Γ xx = V ρ(r)(y2 +z 2 ) dv Γ yy = V ρ(r)(x2 +z 2 ) dv Γ zz = V ρ(r)(x2 +y 2 ) dv Γ xy = Γ yx = V ρ(r)xy dv Γ xz = Γ zx = V ρ(r)xz dv Γ yz = Γ zy = V ρ(r)yz dv Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

12 Rotation of the reference frame The inertia matrix is always defined specifying explicitly or implicitly a point with respect to which it is computed; usually this point is the center of mass of the body, and R b is the body frame. If we consider another frame R k, rotated around the common origin (in this case C) with respect to R b, and the rotation matrix given by R k b, then the inertia matrix is Γ k c = R k b Γb cr b k = Rk b Γb c(r k b )T or, using the usual notation R a b = (Rb a )T Γ k cr k b = Rk b Γb c. Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

13 Principal inertia matrix Often it is appropriate to refer to a privileged reference frame R, implicitly defined as that frame with its origin at the body center of mass and with respect to which the inertia matrix is diagonal, i.e., Γ c = Γ x Γ y 0 ; 0 0 Γ z This frame has the three unit vectors i, j and k, aligned along the so-called principal inertia axes; the matrix Γ c is called principal inertia matrix. Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

14 Parallel axes theorem Assume that we know the inertia matrix Γ with respect to a reference frame with origin in the center of mass C, and we want to compute a new inertia matrix Γ with respect to another frame, with different origin O, but with parallel axes. Γ = Γ ms 2 (t) ] = Γ+m [ t 2 I tt T (ty 2 +t2 z ) t xt y t x t z = Γ+m t x t y (tx 2 +tz) 2 t y t z t x t z t y t z (t 2 x +t 2 y) Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

15 Conclusions Generalized momentum H(t) [ ] [ ] p(t) Mv(t) linear momentum H(t) = = h(t) Γ(t)ω(t) angular momentum Generalized force F(t) [ ] [ ] f(t) F(t) = = τ(t) Ḣ(t) = M v(t) Γ(t)ω(t) + Γ(t) ω(t) Basilio Bona (DAUIN - Politecnico di Torino) Rigid body dynamics October / 16

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