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56 CHAPTER 6. CONSTRAINED RIGID BODY DYNAMICS (J ·(M −1 ·J T · λ+M −1 · Ftotal)−c)i = (A· λ+ b)i = Ai · λ+ bi, the conditions can be reformulated in terms of the i-th block row of matrix A and the i-th block of vector b Ai · λ + bi ≥ 0 (6.12) λi ≥ 0 (Ai · λ + bi) · λi = 0, where A and b are from (6.10) and Ai refers to the block row of A due to constraint i. The last condition in (6.12) says that the components of λi are complementary 2 to the components of Ai · λ + bi = Ci = Ji · a − ci and together with the other two conditions forms the ultimate conditions on λ due to the greater-or-equal constraint i. Equation (6.12) serves the same purpose like (6.10) for equality constraints. Less-or-Equal Constraints Acceleration-level less-or-equal constraint i affecting bodies Ai and Bi is defined as Ji,Ai · aAi + Ji,Bi · aBi ≤ ci, (6.13) where Ji,Ai and Ji,Bi are mi × 6 Jacobian blocks due to the constraint i and the constrained bodies, mi is the dimensionality of the constraint and ci is the right-hand side vector of length mi. The definition pretty much matches the definition of greater-or-equal constraint (6.11), but ≥ is replaced by ≤. Using the notation and following the derivation scheme of greater-or-equal constraints, bodies in the velocity space can accelerate from the hypersurfaces in the directions opposite to the hypersurface normals only. That said, it is required that λi ≤ 0. Also if the bodies are already accelerating to the back of the hypersurface j, ( Ci)j < 0, then the constraint force due to that hypersurface must vanish, that is ( λi)j = 0. That way we obtain the conditions on λ due to the less-or-equal constraint i, expressed as Ai · λ + bi ≤ 0 (6.14) λi ≤ 0 (Ai · λ + bi) · λi = 0, where Ai and bi are from (6.12). Again, the components of λi are complementary to the components of Ai · λ + bi = Ci = Ji · a − ci. 6.2.3 Bounded-Equality Constraints We would like to define a constraint that would behave like an equality constraint whose constraint force magnitude was tracked and when it grew beyond a specified limit the constraint would automatically turn into a corresponding greater-or-equal or less-or-equal constraint. Such a constraint could be used to implement various kinds of motors with limited motor forces so that the constraint would break if the maximum motor force magnitude was exceeded 3 . We will 2 That is ( λi)j = 0 ⇒ (Ai · λ + bi)j = 0 and (Ai · λ + bi)j = 0 ⇒ ( λi)j = 0. 3 Note that such a behavior can not be simulated by treating the constraint as an (unbounded) equality constraint, computing the constraint force and finally clamping the components of λi to specified ranges because other constraints influenced by this constraint would break (constraint forces due to the other constraints were computed with respect to the unclamped λ).
6.2. ACCELERATION-LEVEL CONSTRAINTS 57 define this constraint (as usually) in terms of a constraint force F i c = J T i · λi and conditions on λi multipliers. If the dimensionality of the constraint was one, mi = 1, then the magnitude of the constraint force F i c would be J T i · |( λi)1| and instead of limiting the force magnitude we could limit (the only component of) λi. In the general case we will also limit the (multiple) components of λi, but independently on each other, because the equations would not be linear otherwise. Henceforward we will assume that we are provided with mi-vectors λ lo i ≤ 0 and λ hi i ≥ 0 so that the limits are specified as ( λ lo i )j ≤ ( λi)j ≤ ( λ hi i )j, where 1 ≤ j ≤ mi. According to the notation we already used while deriving conditions for greater-or-equal constraints, bodies in the velocity space will be constrained not to accelerate along the directions of the hypersurface normals due to the corresponding unbounded equality constraint. If the bodies began to accelerate off the hypersurface j due to the j-th constraint DOF in the direction of its normal, a negative ( λi)j would be required to cancel the acceleration. However if ( λi)j dropped below its lower limit ( λ lo i )j, ( λi)j would be clamped and the magnitude of the clamped value would not be big enough to cancel the prohibited acceleration, hence ( Ci)j > 0. Similarly, if the bodies began to accelerate off the hypersurface in the opposite direction, a positive ( λi)j would be required to cancel the acceleration and if ( λi)j exceeded the upper limit ( λ hi i )j, ( λi)j would be clamped and hence ( Ci)j < 0. Since Ci = Ai · λ + bi, these conditions can be written as ( λ lo i )j ≤ 0 ( λ hi i )j ≥ 0 ( λ lo i )j ≤ ( λi)j ≤ ( λ hi i )j ( λi)j = ( λ lo i )j ⇒ (Ai · λ + bi)j ≥ 0 ( λi)j = ( λ hi i )j ⇒ (Ai · λ + bi)j ≤ 0 ( λ lo i )j < ( λi)j < ( λ hi i )j ⇒ (Ai · λ + bi)j = 0, (6.15) where Ai and bi are from (6.10). We can now define the constraint properly. Acceleration-level bounded-equality constraint i affecting bodies Ai and Bi with λi limits λ lo i ≤ 0 and λ hi i ≥ 0 is defined as an equality constraint Ji,Ai · aAi + Ji,Bi · aBi = ci, (6.16) where Ji,Ai and Ji,Bi are mi × 6 Jacobian blocks due to the constraint i and the constrained bodies, mi is the dimensionality of the constraint, ci is the right-hand side vector of length mi and λ lo i and λ hi i are mi-vectors that specify the lower and upper limits of the components of λi due to the constraint. From (6.15) we get therefore if we set λ lo i = 0 and λ hi i (Ai · λ + bi)j > 0 ⇒ ( λi)j = ( λ lo i )j (Ai · λ + bi)j < 0 ⇒ ( λi)j = ( λ hi i )j, greater-or-equal constraint (6.12). Similarly, by setting λ lo i or-equal constraint (6.14). Finally, by setting λ lo i = −−→ +∞ then the bounded equality constraint turns into a = −−→ −∞ and λ hi i = 0 we get a less- = −−→ −∞ and λ hi = −−→ +∞ we get an unbounded i
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Univerzita Karlova v Praze Matemati
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Contents 1 Introduction 1 1.1 Backg
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CONTENTS vii 8 Figure Library 103 8
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x CONTENTS
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2 CHAPTER 1. INTRODUCTION data from
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114 CHAPTER 9. RESULTS Figure 9.2:
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116 CHAPTER 9. RESULTS The produced
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118 CHAPTER 9. RESULTS of optical m
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122 BIBLIOGRAPHY [16] Katsuaki Kawa
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