Dynamics of Mechanical Systems 2009 Part 15 docx

682 Dynamics of Mechanical Systems Then, by substituting from Eqs. (19.13.11) and (19.14.8), we obtain: (19.15.2) Equation (19.15.2) may be written in the compact matrix form as: (19.15.3) where A is a symmetrical (n + 1) × (n + 1) generalized mass array, q is an (n + 1) × 1 column array of generalized coordinates, f is an (n + 1) × 1 column array of inertia forces, and F is an (n + 1) × 1 column array of generalized applied forces. The elements a ᐉ p of A are: (19.5.4) The elements f ᐉ of f are: (19.15.5) The elements f ᐉ of F are: (19.15.6) where the are given by Eq. (19.13.11). 19.16 Redundant Robots Redundant robots have more degrees of freedom than are needed to accomplish a given task. These extra degrees of freedom pose special problems for dynamics analysis. Deci- sions need to be made as to how the superfluous degrees of freedom are to be used. Expressed another way, constraints need to be imposed upon redundant robots for the robots to accomplish a given task. These constraints may take a variety of forms such as minimum joint torques, minimum energy consumption, obstacle avoidance, minimum jerk, or least time of operation. Methods of analysis to accommodate such constraints are still being developed. Difficulties of these analyses have not all been resolved. To avoid these difficulties, early manufacturers of robots have restricted their machines to at most six degrees of freedom; however, without redundancy, the machines cannot accomplish their given tasks in an optimal manner. In this and the following section we will explore some of the issues in the dynamics and control of redundant robot systems. Consider the general system of Figure 19.16.1. Suppose we have a desired movement of the end effector E. For simplicity (and without loss of generality), let us consider E to −+ () −+ () −−++= =… + () =− () mv vq vq I q q Ie qqmgvMF nk k k m kpm p kpm p kmn k m kpn p kpn p ksn k m rsm kqr kpn qp j j k E ll l ll ll ˙˙ ˙ ˙ ˙˙ ˙ ˙ ˙˙ ,, ωω ω ωωω 1 3 0 11 1 Aq f F ˙˙ =+ amvv I p k k m kpm kmn k m k n l lll =+ωω fmvvqI qeI qq k k m kpm p kmn k m kpn p rsm ksn k m kqr kpn pql ll l =− + + () ˙˙ ˙ ˙˙˙ ωω ωωω FmgvM F jj h E ll l l =− + + − 1 3 F E l 0593_C19_fm Page 682 Tuesday, May 7, 2002 9:07 AM Introduction to Robot Dynamics 683 be a single body (that is, suppose E is a gripper firmly holding an object or workpiece). Let P be a typical point, perhaps at an extremity, of E as in Figure 19.16.2. For a given desired motion of E, we will know both the angular velocity of E and the velocity of P. Let us represent the movement of the system by a series of articulation angles γ i (i = 1,…, n), as in Section 19.2. For simplicity (and again without loss of generality) let us assume a fixed base angle α B . Then, we can express the angular velocity of E and the velocity of P in the forms: (19.16.1) where, as before, the n 0m (m = 1, 2, 3) are unit vectors fixed in the inertia frame R, and the v p ᐉ m and ω E ᐉ m are the partial velocity and partial angular velocities, respectively, of P and E. Then, with the movement and P and E being known, we will know the n 0m components of ωω ωω E and V P . These components may then be expressed as: (19.16.2) where the g i (t) (i = 1,…, 6) are known functions of time. Equation (19.16.2) may be written in the compact form: (19.16.3) where γ is a column array of the articulation angles, g is a column array of functions on the right sides of Eq. (19.16.2), and B is a block array of partial angular velocity and partial velocity array components of Eq. (19.16.2). If there are n articulation angles γ i (i = 1,…, n), then γ is an (n × 1) array, g is a (6 × 1) array, and B is a (6 × n) array. B is sometimes called the constraint matrix. The articulation angles are governed by the dynamics equations (Eqs. (19.15.3)): (19.16.4) If there are n articulation angles γ i , Eq. (19.16.4) is equivalent to n scalar equations. Then, Eqs. (19.16.3) and (19.16.4) form a total of (6 + n) equations for the n γ i and the n joint moments M i (i = 1,…, n). Thus, we have (n + 6) equations for 2n unknowns and the system is undetermined. There are (n – 6) fewer equations than needed to uniquely specify the FIGURE 19.16.1 Generic robot system. FIGURE 19.16.2 Representation of an end effector. E P ωω E Em m P pm m v==ωγ γ ll ll ˙˙ nVn 00 and ωγ γ Em m pm m gt V g t m ll ll ˙˙ ,,= () = () = + and 3 123 Bg ˙ γ= AfF ˙˙ γ= + 0593_C19_fm Page 683 Tuesday, May 7, 2002 9:07 AM 684 Dynamics of Mechanical Systems motion of the system. Expressed another way, the robot has (n – 6) more degrees of freedom than needed to uniquely determine the movement of the system. These extra degrees of freedom represent the redundancy in the system, thus we have a redundant robot. To resolve the redundancy, we need (n – 6) additional equations or constraints on the system. As noted earlier, these equations may be obtained in a number of ways, by specifying some of the articulation angles or by specifying some of the joint torques, or a combination thereof, or by imposing more general requirements such as having minimum joint torques or minimum kinetic energy for the system. We will further discuss the form and solutions of the governing equations in the following sections. 19.17 Constraint Equations and Constraint Forces Before we look for solutions of the governing equation let us attempt to obtain further insight into the nature of the equations themselves by considering the following problem. Suppose we have an n-link robot arm with a desired motion of the nth link as in Figure 19.17.1. As before, for simplicity, let us assume that we have a fixed base B. Suppose this system is initially at rest in an arbitrary configuration. Suppose further that, for this discussion, the system is in a weightless and force-free environment and that the moments between adjoining links at the joints are all zero. Imagine further that we are somehow able to move the nth link through its desired motion. Under these rather specialized conditions and with the given motion of the nth link, we could ask the question: What are the resulting motions of the first n – 1 links? That is, in the absence of gravity, with free movement at the joints, and with a specified movement of the last link, what are the movements of the first n – 1 links? To answer this question, consider again the governing dynamical equations (Eqs. (19.16.4)): (19.17.1) where, as before, A is the (n × n) generalized mass array with elements as in Eq. (19.15.4); γ is the array of articulation angles; f is an array of inertia force terms with elements as in Eq. (19.15.5); and F is the array of generalized applied forces with elements F ᐉ as in Eq. (19.15.6). In this rather specialized case with no gravity and with zero joint moments, the F ᐉ have the reduced form: (19.17.2) FIGURE 19.17.1 An n-link robot with a desired motion of the last link. AfF ˙˙ γ= + FF ll = ′ 0593_C19_fm Page 684 Tuesday, May 7, 2002 9:07 AM Introduction to Robot Dynamics 685 where the are generalized forces arising due to the specified motion of the nth link. That is, the are generalized forces due to the applied forces and movements needed to drive the nth link through the specified motion. These forces are constraint forces and the resulting generalized forces of Eqs. (19.17.2) form an array that might be called a generalized constraint force array. The specified motion of the nth link also generates a set of constraint equations of the form of Eq. (19.16.3). That is, (19.17.3) Remarkably, the generalized constraint force array F′ is directly related to the constraint matrix array B by the expression: (19.17.4) where λ is an array of constraint force and moment components and B T is the transpose of B. To see this, suppose a point P of the nth link (perhaps a tool point of an end effector) has a specified motion (see Figure 19.17.2). Suppose further that the angular motion of the link L n is specified. Suppose still further that the specified motions of P and L n are given in the form of velocities or in a form such that the velocity of P and the angular velocity of L n may be determined. Then, the velocity of G n , the mass center of L n , will be known or can readily be determined. Let these velocities be expressed in the forms: (19.17.5) and (19.17.6) where νν νν (t) and ΩΩ ΩΩ (t) are known (specified) vector functions of time. Referring to the unit vectors n 0m of R it is convenient to express V n and ωω ωω n in the component forms: (19.17.7) FIGURE 19.17.2 Specified motion of link L n . ′ F l ′ F l ′ F l Bg ˙ γ= ′ =FB T λ R G n n tVV== () ν R L n n tωωωωΩΩ== () Vn n nm G mnm L m V nn == ll ll ˙˙ γωγ 00 and ωω R n n n L V P ω G P n n n 03 02 01 0593_C19_fm Page 685 Tuesday, May 7, 2002 9:07 AM 686 Dynamics of Mechanical Systems (Observe that this is different than our usual notation of V n ᐉ m n 0m and ω n ᐉ m n 0m . The change is introduced to simplify the matrix analysis of the following paragraphs.) In scalar form, Eqs. (19.17.5) and (19.17.6) become: (19.17.8) where ν m (t) and Ω m (t), the n 0m components of νν νν and ΩΩ ΩΩ , are specified functions of time. The expressions in Eq. (19.17.8) are constraint equations in the form of Eqs. (19.16.3) and (19.17.3). Specifically, the constraint matrix B is a (6 × n) array whose elements are the partial velocity and partial angular velocity components. That is, (19.17.9) Then, by comparison of Eqs. (19.17.3), (19.17.8), and (19.17.4), we see that the array g is: (19.17.10) Next, let the constraining force system on L n be represented by an equivalent force system consisting of a force passing through G n together with a couple with torque as in Figure 19.17.3. Then, the generalized constraint for force for is (see Eq. (11.5.7)): (19.17.11) FIGURE 19.17.3 Equivalent constraint force system on link L n . L G n n M ' n F ' n Vt tm m G mm L m nn ll ll ˙˙ ,,γν ωγ= () = () =and Ω 123 B VVV V VVV V VVV V GGG n G GGG n G GGG n G LLL n L LLL n L LLL n L nnn n nnn n nnn n nnn n nnn n nnn n =       11 12 13 1 21 22 23 2 31 32 33 3 21 12 13 1 21 22 13 2 32 32 33 3 L L L L L L ωωω ω ωωω ω ωωω ω               gt t t t t t t () = () () () () () ()                     ν ν ν 1 2 3 1 2 3 Ω Ω Ω ′ F n ′ M n ′ F l ˙ γ l ′ = ′ + ′ =… () FFV M n nm m G nm m L nn ll l lω 1, , 0593_C19_fm Page 686 Tuesday, May 7, 2002 9:07 AM Introduction to Robot Dynamics 687 where the and are the n 0m components of and . Let λ be the (6 × 1) column array of these components. That is, (19.17.12) Observe from Eq. (19.17.9) that the transpose of B is: (19.17.13) Then, by inspection of Eq. (19.17.11) and by comparison with Eqs. (19.17.12) and (19.17.13), we see that: (19.17.14) thus establishing Eq. (19.17.4). In view of Eq. (19.17.14) the governing equations, Eqs. (19.17.1) and (19.17.3), may be written as: (19.17.15) and (19.17.16) We will consider solutions to these equations in the following section. 19.18 Governing Equation Reduction and Solution: Use of Orthogonal Complement Arrays Recall that Eqs. (19.17.15) and (19.17.16) are the governing equations for an n-link robot whose nth link is driven with a prescribed motion. The joints are moment free, and no gravitational or externally applied forces are present, other than the constraint forces ′ F nm ′ M nm ′ F n ′ M n λ= ′ ′ ′ ′ ′ ′                     F F F M M M n n n n n n 1 2 3 1 2 3 B VVV VVV VVV VVV T GGGLLL GGGLLL GGGLLL n G n G n G n L n L n L nnnnnn nnnnnn nnnnnn n n nnnn =      11 21 31 11 21 31 12 22 32 12 22 32 13 23 33 13 23 33 123123 ωωω ωωω ωωω ωωω MMMMMM            ′ =FB T λ AfB T ˙˙ γλ=+ Bg ˙ γ= 0593_C19_fm Page 687 Tuesday, May 7, 2002 9:07 AM 688 Dynamics of Mechanical Systems needed to drive the nth link through its desired motion. Also, the base is fixed. Under these specialized conditions, Eq. (19.17.15) is equivalent to n scalar equations involving the n articulation angles γ i (i = 1,…, n) and the six constraint (or driving) force and moment components. Correspondingly, Eq. (19.17.16) is equivalent to six scalar constraint equations involving the articulation angles. Therefore, taken together, Eqs. (19.17.15) and (19.17.16) are equivalent to (n + 6) scalar equations for the (n + 6) variables consisting of the n articulation angles and the six driving force and moment components. Equations (19.17.15) and (19.17.16) are nonlinear, coupled differential–algebraic equa- tions. As such, we cannot expect to obtain an analytical solution. Instead, we are left to seek numerical solutions. Generally, we are more interested in knowing the system motion than in knowing the constraint force and moment components. Hence, the question arising is can we reduce and thus simplify the equations by eliminating the λ array? To answer this question, observe that λ may be eliminated if we could find an (n – 6) × n array (say, C T ), with rank (n – 6) such that C T B T is zero. That is, λ may be eliminated from Eq. (19.17.15) by premultiplying by C T if C T B T is zero. Equation (19.17.15) then becomes: (19.18.1) Equation (19.17.16) may then be differentiated and cast into the same format as: (19.18.2) Equations (19.18.1) and (19.18.2) may be combined into a single matrix equation of the form: (19.18.3) where Q is an (n × n) array which has the partitioned form: (19.18.4) and h is an (n × 1) column array with the partitioned form: (19.18.5) Then, with Q being nonsingular, Eq. (19.18.3) may be solved for the as: (19.18.6) Equation (19.18.6) is equivalent to n scalar differential equations for the n articulation angles. They are in a format that is ideally suited for numerical integration using any of a number of numerical integration algorithms (solvers).* * A discussion of such software is beyond our scope here, but interested readers may want to consider numerical analysis writings, such as References 19.2 to 19.5, for information about such software. CA Cf TT ˙˙ γ= BgB ˙˙ ˙ ˙ ˙ γγ=− Qh ˙˙ γ= Q CA B T =       h Cf gB T = −       ˙ ˙ ˙ γ ˙˙ γ ˙˙ γ= − Qh 1 0593_C19_fm Page 688 Tuesday, May 7, 2002 9:07 AM Introduction to Robot Dynamics 689 The key to developing Eq. (19.18.6) is in obtaining the (n – 6) × n array C T with rank (n – 6) such that: (19.18.7) Taking the transpose of Eq. (19.18.7), we have: (19.18.8) In this context, C is often called an orthogonal complement of B. One way of obtaining an orthogonal complement of B is to use a rather ingenious zero- eigenvalues theorem as developed by Walton and Steeves [19.6]. In this procedure, the (6 × n) array B is premultiplied by its transpose B T , forming an (n × n) symmetric array B T B of rank 6. Then, in the eigenvalue problem: (19.18.9) there are (n – 6) zero eigenvalues µ. Let x i (i = 1,…, n – 6) be the eigenvectors x associated with the zero eigenvalues. Next, let C be an n × (n – 6) array whose columns are the (n – 6) eigenvectors x i , associated with the zero eigenvalues. Then, with the eigenvalues µ being zero, we have: (19.18.10) Finally, by premultiplying by C T we have: (19.18.11) Thus, C is the desired orthogonal complement array. Once C is known, the arrays Q and h may be immediately calculated from Eqs. (19.18.4) and (19.18.5), and the articulation angles γ i may then be obtained by integrating Eq. (19.18.6). Observe that Eq. (19.18.3) represents a reduced form of the governing equations obtained by eliminating the constraint force and moment component array λ from Eq. (19.17.15). If one is interested in knowing these constraint force and moment components, they may be obtained by back substitution into Eq. (19.17.15). Specifically, Eq. (19.17.15) may be solved for λ as: (19.18.12) 19.19 Discussion, Concluding Remarks, and Closure The foregoing analysis, although somewhat specialized by our simplifying assumptions, is nevertheless typical of the kinds of analyses used when studying the dynamics of large CB TT = 0 BC = 0 BBx x T =µ BBC T = 0 C B BC BC BC BC BC C B TT T TT = () = () = == 000 00 2 or or or and λγ= () − () − BB A f T 1 ˙˙ 0593_C19_fm Page 689 Tuesday, May 7, 2002 9:07 AM 690 Dynamics of Mechanical Systems systems. For an actual robot, the analysis would of course be more detailed, but the basic principles would be the same. In robot hardware design, analysts are concerned with end effector mechanisms, increased mobility, inverse kinematic problems (determining the articulation angles to obtain a desired end effector movement), multiple arm systems, control problems, location sensing, and effects of flexibility and compliance. The immediate foregoing analysis illustrates a procedure for working with constraint equations. To discuss this further, consider a matrix form of Kane’s equations for a spec- ified-motion constrained system (a so-called acatastatic system [19.7]): (19.19.1) where F is a column array of generalized applied forces, F* is the corresponding array of generalized inertia forces, and F′ is an array of constraint forces. The constraint force array arises as a result of the specified motion of the end effector. The constraint force array is actually a specialized applied force array due to the forces needed to obtain the specified end effector motion. In terms of these force components we have seen that F′ has the form: (19.19.2) where B is the array of constraint equation coefficients and λ is the array of constraint force and moment components. In many cases, the values of the constraint force and moment components are not of interest. In those cases, their array λ may be eliminated from the analysis by premultiplying by the transpose of an orthogonal complement C of B. That is, because BC and C T B T are zero, Eq. (19.19.1) may be reduced to the form: (19.19.3) Equations (19.19.1) and (19.19.3) may be depicted graphically as in Figure 19.19.1, where F, F*, and F′ form a generalized force triangle representing Eq. (19.19.1). The projection of F and F* perpendicular to F′, in the form of C T F and C T F* as shown, represents Eq. (19.19.3). In this sense, the system may be viewed as being able to move in directions orthogonal to the constraints. As we saw in the foregoing analysis, once Eq. (19.19.3) is solved we can use the solution with back substitution into Eq. (19.19.1) to obtain the force and moment components of λ, if they are desired. The references provide a source for more specific information and analysis of robot systems. As with many other current technologies, robotics is rapidly developing. It is beyond our scope, however, to attempt to be more detailed about specific systems than we have been. Our objective has been simply to present an application of our dynamical procedures. Biomechanics represents another and somewhat related application area that we will discuss in the following chapter. FIGURE 19.19.1 Generalized force triangle. F F ' F C F * C F * T T FF F++ ′ = * 0 ′ =FB T λ CF CF TT += * 0 0593_C19_fm Page 690 Tuesday, May 7, 2002 9:07 AM Introduction to Robot Dynamics 691 References 19.1. Kane, T. R., Analytical Elements of Mechanics, Vol. 1, Academic Press, New York, 1959, p. 128. 19.2. Champion, E. R., Jr., Numerical Methods for Engineering Applications, Marcel Dekker, New York, 1993. 19.3. Hornbeck, R. W., Numerical Methods, Quantum Publishers, New York, 1975. 19.4. Griffiths, D. V., and Smith, I. M., Numerical Methods for Engineers, CRC Press, Boca Raton, FL, 1991. 19.5. Rice, J. R., Numerical Methods, Software, and Analysis, McGraw-Hill, New York, 1993. 19.6. Walton, W. C., Jr., and Steeves, E. C., A New Matrix Theorem and Its Application for Estab- lishing Independent Coordinates for Complex Dynamical Systems with Constraints, NASA Technical Report TR-326, 1986. 19.7. Pars, L. A., A Treatise on Analytical Dynamics, Ox Box Press, Woodbridge, CT, 1979, p. 24. 19.8. Angeles, J., Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms, Spring- er-Verlag, New York, 1997. 19.9. Craig, J. J., Introduction to Robotics: Mechanics and Control, Addison-Wesley, Reading, MA, 1989. 19.10. Duffy, J., Analysis of Mechanisms and Robot Manipulators, Edward Arnold, London, 1980. 19.11. Engelberger, J. F., Management and Application of Industrial Robots, American Management Association, New York, 1980. 19.12. Featherstone, R., Robot Dynamics Algorithms, Kluwer Academic, Dordrecht, 1987. 19.13. Koivo, A. J., Fundamentals for Control of Robotic Manipulators, Wiley, New York, 1989. 19.14. Mason, M. T., and Salisbury, J. K., Robot Hands and the Mechanics of Manipulation, MIT Press, Cambridge, MA, 1985. 19.15. Murray, R. M., Li, Z., and Sastry, S. S., A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1993. 19.16. Nakamura, Y., Advanced Robotics: Redundancy and Optimization, Addison-Wesley, Reading, MA, 1991. 19.17. Paul, R. P., Robot Manipulators, Mathematics, Programming, and Control, MIT Press, Cambridge, MA, 1981. 19.18. Spong, M. W., and Vidyasagar, M. M., Dynamics and Control of Robot Manipulators, Wiley, New York, 1989. Problems Section 19.2 Geometry, Configuration, and Degrees of Freedom P19.2.1: Refer to the simple planar robots shown in Figures P19.2.1a and b consisting of two- and three-link arms, respectively, and consider the following questions: 1. How many degrees of freedom does each system have? 2. What are the hinge inclination angles for these systems? 3. From an analysis perspective, what are the relative advantages and disadvan- tages of these systems? 4. From an applications perspective, what are the relative advantages and disad- vantages of these systems? P19.2.2: Consider the simple end effector, or gripper, shown in Figure P19.2.2. If this end effector were attached to the extremities of the systems of Problem P19.2.1, what would 0593_C19_fm Page 691 Tuesday, May 7, 2002 9:07 AM [...]... 7, 2002 9:07 AM 692 a Two-Link System Dynamics of Mechanical Systems b Three-Link System FIGURE P19.2.1 Two- and three-link planar robots FIGURE P19.2.2 An end effector be the resulting degrees of freedom of each system? Would the presence of the end effector affect the relative advantages and disadvantages of each system? P19.2.3: Consider a robot arm consisting of four identical links each having length... 20.3.1 then provides a listing and labeling of the various links and segments of the model By inspection of Figure 20.3.1, we then immediately obtain the lower body array L(K) as follows: K: L(K): 0 0 1 0 2 1 3 2 4 3 5 4 6 5 7 3 8 7 9 3 10 9 11 10 12 1 13 12 14 13 15 1 16 15 17 16 0593_C20_fm Page 704 Tuesday, May 7, 2002 9:14 AM 704 Dynamics of Mechanical Systems 8 4 3 9 5 2 10 6 1 11 7 12 13 16 14... typical adjoining bodies n j1 n k2 n k1 15 1 0 0 0 0 0 16 15 1 0 0 0 0 17 16 15 1 0 0 0 0593_C20_fm Page 706 Tuesday, May 7, 2002 9:14 AM 706 Dynamics of Mechanical Systems Bryan angles, there is a singularity where βk is either 90° or 270° From a computational (or numerical analysis) viewpoint, it is necessary to avoid these singularities if the range of motion of the bodies allows them to occur As... 20.5.1 In developing the kinematics of this system, we are (as with other multibody systems) interested in knowing the position, velocity, and acceleration of the mass center of each body of the system and the angular velocity and angular acceleration of each body In addition, depending upon the application, we may be interested in knowing the kinematics of other parts of the model as well For example,... 14 15 S04 S04 16 17 18 S05 19 20 21 S03 S03 22 23 24 S07 25 26 27 S03 S03 S03 28 29 30 Nonzero Partial Angular Velocity Vector Components () for the Model of Figure 20.5.1 TABLE 20.5.2 S09 S09 31 32 33 S010 34 35 36 S01 S01 S01 37 38 39 S012 S012 40 41 42 S013 43 44 45 S01 S01 S01 46 47 48 S 015 S 015 49 50 51 S016 52 53 54 0593_C20_fm Page 712 Tuesday, May 7, 2002 9:14 AM 712 Dynamics of Mechanical Systems. .. challenging to dynamics analysts As such they provide motivation for the development of modeling and analysis procedures As with other mechanical systems, the objective is to find simple models that are sufficiently representative of the physical system to provide useful information To this end, Figure 20.1.1 shows a typical whole-body or gross-motion model of the human body It consists of a series of bodies... multibody systems Those same methods are applicable here Indeed, obtaining 0593_C20_fm Page 707 Tuesday, May 7, 2002 9:14 AM Application with Biosystems, Human Body Dynamics 707 8 4 3 9 5 2 10 6 1 11 7 12 13 16 14 FIGURE 20.4.1 Human body model 15 17 the kinematics of human body models has been a motivating factor in the development of many of these methods As before, we seek to find for each body of the... Equations of Motion P19 .15. 1: Using Eqs (19.13.11), (19.14.8), and (19 .15. 1), verify Eqs (19 .15. 2) through (19 .15. 6) P19.5.2: See Problems P19.13.1 and P19.14.2 Determine the equations of motion of the simple three-link robot shown in Figure P19.13.1 (Use the same data and parameters as in Problems P19.13.1 and P19.14.2.) Sections 19.16 Redundant Robots P19.16.1: Consider the system of three identical pin-connected... FIGURE P19.16.1 A three-link system with prescribed motion of the extremity γ1 O X P19.16.2: See Problem 19.16.1 Repeat Problem P19.16.1 if E is required to move on a circle of radius r with center C (xO, yO) where r, xO, and yO are compatible with the geometry of the system 0593_C19_fm Page 700 Tuesday, May 7, 2002 9:07 AM 700 Dynamics of Mechanical Systems P19.16.3: See Problems P19.16.1 and P19.16.2... the end of the chapter Unlike a robot, a biosystem — specifically, the human body — is more complex and less well defined The human form is not composed of simple geometrical shapes, and, even if the body is studied in small parts such as with arms, hands, or fingers, the analysis is not simple Indeed, even the location of the mass center of elemental parts is imprecise, and comprehensive analyses of joint . extremities of the systems of Problem P19.2.1, what would 0593_C19_fm Page 691 Tuesday, May 7, 2002 9:07 AM 692 Dynamics of Mechanical Systems be the resulting degrees of freedom of each system?. AM Introduction to Robot Dynamics 699 Section 19 .15 Dynamics: Equations of Motion P19 .15. 1: Using Eqs. (19.13.11), (19.14.8), and (19 .15. 1), verify Eqs. (19 .15. 2) through (19 .15. 6). P19.5.2: See Problems. 682 Dynamics of Mechanical Systems Then, by substituting from Eqs. (19.13.11) and (19.14.8), we obtain: (19 .15. 2) Equation (19 .15. 2) may be written in the compact matrix form as: (19 .15. 3) where