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46 CHAPTER 5. RIGID BODY DYNAMICS Second Law for particles and rigid bodies can thus be imagined as special particles with timevarying masses M(t) that move in R 6 . The formulation of rigid body dynamics with mass matrices will become very useful when the equations for constrained rigid body dynamics are developed because we will be able to reuse derivations from constrained particle dynamics that were formulated in terms of mass matrices for particles. 5.2.4 Numerical Issues If we actually tried to simulate a rigid body, say according to (5.18), we would experience a numerical drift which would mostly affect the representation of the body orientation R(t). The problem is that rigid bodies have 3 rotational degrees of freedom but rotation matrices R(t) have 3 × 3 DOFs and the extra DOFs have to be neutralized by 6 additional conditions on the matrix rows, known as the orthogonality conditions. While updating R(t), orthogonality conditions are violated and the matrix must be “orthogonalized” in order to represent a rotation again. Whenever the normalization takes place, rigid body state is changed directly, which makes it a new initial state and the ODE solver must be restarted. There is, however, a better way to represent body orientations — by the use of unit quaternions, represented as [scalar = s, −−−−→ vector = (v1, v2, v3)] pairs. Quaternions, unlike rotation matrices, have 4 DOFs and the extra DOF is neutralized by a condition that requests that the quaternion must have a unit length. Since there is only one constrained DOF, orientations represented by unit quaternions experience much less numerical drift than rotation matrices and the stabilization of the corresponding ODE is much simpler (corresponds to the normalization of the quaternion). We will assume that the reader is acquainted with quaternions and the representation of rotations by unit quaternions and present a modified equation of motion for a single rigid body straight off (one might read [1] to get introductory level information on quaternions). A variant of (5.18) with a different block row due to orientation representation by a quaternion is shown ∂ y(t) ∂t = ∂ ∂t (x(t), q(t), P (t), L(t)) = (v(t), 1 2 · [0, ω(t)] · q(t), Ftotal(t), τtotal(t)), where q(t) = [cos(α/2), sin(α/2) · v] is a unit quaternion that represents the body orientation by a rotation about axis v by angle α. To utilize previous equations expressed in terms of R(t), one might consider R(t) as an additional auxiliary quantity, computed from the quaternion q(t). Derivations can be found in [10]. 5.3 Abstraction In this section we will cover the abstraction and representation of rigid bodies and external forces that may act upon them. The abstractions generalize and built on top of the previous abstractions of particles and force objects described in the particle dynamics chapter. We already know that the simulation of rigid bodies and particles corresponds to the numerical solving of the corresponding equations of motion, which are coupled ODEs. Since we have got multiple forms of equations of motion to choose from, we have to decide what variant of the equation of motion is to be used to drive the simulation and then implement the functions required by the ODE solver (see page 13) accordingly. The simulation of rigid bodies conceptually follows
5.3. ABSTRACTION 47 the same scheme like the simulation of particles and the only thing which makes it harder are the extra dimensions due to body orientations. We will use the momentum-based formulation of the equation of motion (5.18) to simulate the bodies. This decision dictates what the layout of the body state vector y(t) is and what properties are recorded in the body state (equation (5.17)). Obviously, vector y(t) will be a part of our rigid body representation. The right-hand side of (5.18) tells us what quantities have to be processed when the body state derivative is requested to be evaluated by the ODE solver. Since the quantities are defined in terms of auxiliary quantities v(t), ω(t), I(t) and I −1 (t), derived from the rigid body state, constant quantities Ibody, I −1 body , M and M −1 , characterizing the rigid body mass properties, and computed quantities Ftotal(t), τtotal(t), storing the total external force and torque acting on the body at time t, all these quantities should be represented as well. Let us focus at Ftotal(t) and τtotal(t) quantities. When these quantities were defined (equations (5.7), (5.9)), we were conceptually working with a rigid body made of particles, correlated the positions where the forces were exerted with the positions of particles and defined a torque according to the position of the particle. Since we can imagine that there happens to be a particle at any world space position we are interested in, we can interpret τi(t) from (5.7) as a torque due to an external force Fi(t) exerted on the body at the world space position ri(t) and hence can write n Ftotal(t) = Fi(t) n n τtotal(t) = τi(t) = (ri(t) − x(t)) × i=1 i=1 i=1 Fi(t), where Fi(t) is an external force exerted at point ri(t) on the body. This allows to implement Ftotal(t) and τtotal(t) as a force and torque accumulators and handle the contributions to the accumulators in terms of apply force(r, F ) function that adds the exerted force F to the body’s Ftotal(t) accumulator and the corresponding torque (r − x(t)) × F due to F exerted at r to the τtotal(t) accumulator. The contributions are represented by force objects that provide functions to exert the represented forces and torques in terms of apply force function called on affected bodies.
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Univerzita Karlova v Praze Matemati
Page 5 and 6: Contents 1 Introduction 1 1.1 Backg
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122 BIBLIOGRAPHY [16] Katsuaki Kawa
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