Advanced Strategies For Robot Manipulators Part 2 docx

The Comparative Assessment of Modelling and Control of Mechanical Manipulator 21 0 0.5 1 1.5 2 2.5 -20 0 20 40 60 80 100 120 Time (s) Angular Position-Joint 1 (degree) case 1 case 2 case 3 case 4 0 0.5 1 1.5 2 2.5 20 40 60 80 100 120 140 160 180 Time (s) Angular Position-Joint 2 (degree) case 1 case 2 case 3 case 4 Fig. 12. Angular positions of joints 0 0.5 1 1.5 2 2.5 -4 -3 -2 -1 0 1 2 Time (s) Angular Velocity-Joint 1 (rad/s) case 1 case 2 case 3 case 4 0 0.5 1 1.5 2 2.5 -6 -4 -2 0 2 4 Time (s) Angular Velocity-Joint 2 (rad/s) case 1 case 2 case 3 case 4 Fig. 13. Angular velocities of joints 0 0.5 1 1.5 2 2.5 -50 0 50 Time (s) Torque Motor 1 (N-m) case 1 case 2 case 3 case 4 upper bound lower bound 0 0.5 1 1.5 2 2.5 -60 -40 -20 0 20 40 60 Time (s) Torque Motor 2 (N-m) case 1 case 2 case 3 case 4 upper bound lower bound Fig. 14. Torques of motors Advanced Strategies for Robot Manipulators 22 -1 -0.5 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 3 x(m) y(m) case 1 case 2 case 3 case 4 Fig. 15. End effector trajectory in XY plane 3.2.3 Maximum payload determination In this case, the maximum payload of flexible mobile manipulator will be calculated and corresponding optimal trajectory at point-to-point motion will be illustrated for different values of W. Payload paths for these cases are shown in Fig. 16. Fig. 17 shows the robot configuration for the first and last cases. Also, he computed torques for these cases are plotted in Fig.18. As it can be seen, increasing W causes to increase oscillatory behavior of the systems that results to reduce the maximum dynamic payload as shown in Table 3. Case 1 2 3 4 W 1 400 600 800 maxp m 8.4 7.9 7.5 6.3 Table 3. The values of W and corresponding calculated maximum payloads -1 -0.5 0 0.5 1 1.5 2 2.5 0.5 1 1.5 2 2.5 3 x(m) y(m) case 1 case 2 case 3 case 4 Fig. 16. End effector trajectory in XY plane The Comparative Assessment of Modelling and Control of Mechanical Manipulator 23 -1 0 1 2 3 0.5 1 1.5 2 2.5 3 x(m) y(m) Robot Configuration Case 1 -1 0 1 2 3 0.5 1 1.5 2 2.5 3 x(m) y(m) Robot Configuration Case 4 Fig. 17. Robot Configuration 0 0.5 1 1.5 2 2.5 -50 0 50 Time (s) Torque Motor 1&2 (N-m) Motor 1 Motor 2 Case 1 0 0.5 1 1.5 2 2.5 -30 -20 -10 0 10 20 30 40 Time (s) Torque Motor 1&2 (N-m) Motor 1 Motor 2 case 4 Fig. 18. Torques of motors 4. Conclusion In this chapter, modelling and control of mechanical manipulator had been studied. First, kinematic and dynamic modelling of flexible link, flexible joint and mobile manipulators have been considered. Then, optimal control of a flexible mobile manipulator in point-to- point motion had been formulated based on the open-loop optimal control approach. The first objective of the chapter is to state the dynamic optimization problem under a quite generalized form in order to be applied to a variety of situations with any guess objective functions for the optimality solution. The second objective is consisting in developing the method for optimizing the applicable case studies, which results. Using assumed mode and finite element methods oscillatory behavior of he mobile robotic manipulators had been described. The model equations had been verified for a two-link manipulator, and the model responses had been discussed. Then, joint flexibility had been added to the system and obtained model had been simulated. After that, an efficient solution on the basis of TPBVP solution had been proposed to path optimization – maximum payload determination in order to achieve the predefined objective. The solving strategy makes it possible to get any guess objective functions for the optimality solution. Attaining the minimum effort trajectory along with bounding the obtained velocity magnitude had been chosen at the application example. The obtained results illustrate the Advanced Strategies for Robot Manipulators 24 power and efficiency of the method to overcome the highly nonlinearity nature of the optimization problem which with other methods, it may be very difficult or impossible. Highlighting the main contribution of the chapter can be presented as: • The proposed approach can be adapted to any general serial manipulator including both non-redundant and redundant systems with link flexibility and base mobility. • In this approach the nonholonomic constraints do not appear in TPBVP directly, unlike the method given in (Mohri et al. 2001; Furuno et al. 2003). • This approach allows completely nonlinear states and control constraints treated without any simplifications. • The obtained results illustrate the power and efficiency of the method to overcome the high nonlinearity nature of the optimization problem, which with other methods, it may be very difficult or impossible. • In this method, boundary conditions are satisfied exactly, while the results obtained by methods such as Iterative Linear Programming (ILP) have a considerable error in final time (Ghariblu & Korayem, 2006). • In this method, designer is able to choose the most appropriate path among various optimal paths by considering the proper penalty matrices. The optimal trajectory and corresponding input control obtained using this method can be used as a reference signal and feed forward command in the closed-loop control of such manipulators. 5. References Bertolazzi E.; Biral F. & Da Lio M. (2005). Symbolic–numeric indirect method for solving optimal control problems for large multibody systems, Multibody System Dynamics, Vol. 13, No. 2, pp. 233-252 Bessonnet G. & Chessé S. (2005). Optimal dynamics of actuated kinematic chains, Part 2: Problem statements and computational aspects, European J. of Mechanics A/Solids, Vol. 24, pp. 472–490 Bloch A. M. (2003). Nonholonomic mechanics and control. Springer, New York Bock H. G. & Plitt K. J. (1984). A multiple shooting algorithm for direct solution of optimal control problems, Proc. 9th IFAC World Congress, pp. 242–247 Callies R. & Rentrop P. (2008). Optimal control of rigid-link manipulators by indirect methods, GAMM-Mitt., Vol 31, No. 1, pp. 27 – 58 Chen W. (2001). Dynamic modelling of multi-link flexible robotic manipulators, Computers and Structures, Vol. 79, (2), pp. 183–195 Furuno S.; Yamamoto M. & Mohri A. (2003). Trajectory planning of mobile manipulator with stability considerations, Proc. IEEE Int. Conf. on Robotics and Automation, pp. 3403-3408 Gariblu H. & Korayem M. H. (2006). Trajectory Optimization of Flexible Mobile Manipulators, Robotica, Vol. 24, No. 3, pp. 333-335 Green A. & Sasiadek J.Z. (2004). Dynamics and Trajectory Tracking Control of a Two-Link Robot Manipulator, Journal of Vibration and Control, Vol. 10, No. 10, pp. 1415–1440 Hargraves C. R. & Paris S. W. (1987). Direct trajectory optimization using nonlinear programming and collocation, AIAA J. Guidance, Vol. 10, No. 4, pp. 338–342, 1987. The Comparative Assessment of Modelling and Control of Mechanical Manipulator 25 Korayem M. H. & Ghariblu H. (2004). Analysis of wheeled mobile flexible manipulator dynamic motions with maximum load carrying capacities, Robotics and Autonomous Systems, Vol. 48, No. 2-3, pp. 63-76 Korayem M.H. & Rahimi Nohooji H. (2008). Trajectory optimization of flexible mobile manipulators using open-loop optimal control method, LNAI, Springer-Verlag Berlin Heidelberg, Vol. 5314, Part 1, pp. 54–63. Korayem M. H.; Haghpanahi M. ; Rahimi H. N. & Nikoobin A. (2009). Finite element method and optimal control theory for path planning of elastic manipulators, New Advanced in Intelligent Decision Technology., SCI, Springer-Verlag Berlin Heidelberg, Vol. 199, pp. 107-116 Korayem M. H.; Rahimi H. N. & Nikoobin A. (2009). Analysis of Four Wheeled Flexible Joint Robotic Arms with Application on Optimal Motion Design, New Advanced in Intelligent Decision Technology., SCI, Springer-Verlag Berlin Heidelberg, Vol. 199, pp. 117-126 Mohamed Z. & Tokhi M.O. (2004). Command shaping techniques for vibration control of a flexible robot manipulator, Mechatronics, Vol. 14, pp. 69–90. Mohri A.; Furuno S. & Yamamoto M. (2001). Trajectory planning of mobile manipulator with end-effector's specified path, Proc. IEEE Int. Conf. on Intelligent Robots and systems, pp. 2264-2269 Papadopoulos E. & Rey, D. (1996). A New measure of tip over stability margin for mobile manipulators, Proc. IEEE Int. Conference on Robotics and Automation, pp. 3111-3116 Papadopoulos E. & Gonthier, Y. (1999) A framework for large-force task planning of mobile redundant manipulators, J. of Robotic Systems, Vol. 16, No. 3, pp. 151-162 Papadopoulos E.; Poulakakis I. & Papadimitriou I. (2002). On path planning and obstacle avoidance for nonholonomic platforms with manipulators: A polynomial approach, Int. J. of Robotics Research, Vol. 21, No. 4, pp. 367-383 Rahimi H. N.; Korayem M. H. & Nikoobin A. (2009). Optimal Motion Planning of Manipulators with Elastic Links and Joints in Generalized Point-to-Point Task, ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE), Vol. 7, Part B, 33rd Mechanisms and Robotics Conference, pp 1167-1174, San Diego, CA, USA Sentinella M. R. & Casalino L. (2006). Genetic algorithm and indirect method coupling for low-thrust trajectory optimization, 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, California Seraji H., "A unified approach to motion control of mobile manipulators, Int. J. of Robotic Research, Vol. 17, No. 12, pp.107–118 (1998). Shampine L. F.; Reichelt M. W. & Kierzenka J. Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c, available at http://www.mathworks.com/bvp tutorial Sheng Ge Xin & Qun Chen Li. (2006). Optimal motion planning for nonholonomic systems using genetic algorithm with wavelet approximation, Applied Mathematics and Computation, Vol. 180, pp. 76–85 Subudhi B. & Morris A.S. (2002). Dynamic Modelling, Simulation and Control of a Manipulator with Flexible Links and Joints, Robotics and Autonomous Systems, Vol. 41, pp. 257–270 Advanced Strategies for Robot Manipulators 26 Usoro P.B.; Nadira R., Mahil S. S. (1986). A finite element/Lagrange approach to modelling lightweight flexible manipulators, J. of Dynamics Systems, Measurement, and Control, Vol. 108, pp.198–205 Wachter A. & Biegler L. T. (2006). On the implementation of an interior-point filter line- search algorithm for large-scale nonlinear programming, Mathematical Programming, Vol. 106, No. 1, pp. 25–57 Yue S., Tso S. K. & Xu W. L. (2001). Maximum dynamic payload trajectory for flexible robot manipulators with kinematic redundancy, Mechanism and Machine Theory 36, 785– 800 Zhang C. X. & Yu Y. Q. (2004). Dynamic analysis of planar cooperative manipulators with link flexibility, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 126, pp. 442–448 2 Hyper Redundant Manipulators Ivanescu Mircea and Cojocaru Dorian University of Craiova Romania 1. Introduction Modern industrial robots are mostly (human) arm-inspired mechanisms with serially arranged discrete links. When it comes to industrial environment where the workspace is structured and predefined this kind of structure is fine. This type of robots are placed in carefully controlled environments and kept away from human and their world. When it comes to robots that must interact with the natural world, it needs to be able to solve the same problems that animals do. The rigid structure of traditional robots limit their ability to maneuver and in small spaces and congested environments, and to adapt to variations in their environmental contact conditions. For improving the adaptability and versatility of robots, recently there has been interest and research in “soft” robots. In particular, several research groups are investigating robots based on continuous body “continuum” structure. If a robot’s body is soft and/or continuously bendable it might emulate a snake or an eel with an undulating locomotion (Walker & Carreras, 2006). An ideal tentacle manipulator is a non-conventional robotic arm with an infinite mobility. It has the capability of takeing sophisticated shapes and of achieving any position and orientation in a 3D space. Behavior similar to biological trunks, tentacles, or snakes may be exhibited by continuum or hyper-redundant robot manipulators (Walker et al., 2005). Hence these manipulators are extremely dexterous, compliant, and are capable of dynamic adaptive manipulation in unstructured environments, continuum robot manipulators do not have rigid joints unlike traditional rigid-link robot manipulators. The movement of the continuum robot mechanisms is generated by bending continuously along their length to produce a sequence of smooth curves. This contrasts with discrete robot devices, which generate movement at independent joints separated by supporting links. The snake-arm robots and elephant’s trunk robots are also described as continuum robots, although these descriptions are restrictive in their definitions and cannot be applied to all snake-arm robots (Hirose, 1993). A continuum robot is a continuously curving manipulator, much like the arm of an octopus (Cowan & Walker, 2008). An elephant’s trunk robot is a good descriptor of a continuum robot (Hutchinson, S.; Hager et al., 1996). The elephant’s trunk robot has been generally associated with an arm manipulation – an entire arm used to grasp and manipulate objects, the same way that an elephant would pick up a ball. As the best term for this class of robots has not been agreed upon, this is still an emerging issue. Snake-arm robots are often used in association with another device meant to introduce the snake-arm into the confined space. However, the development of high-performance control algorithms for these manipulators is quite a challenge, due to their unique design and the high degree of uncertainty in their Advanced Strategies for Robot Manipulators 28 dynamic models. The great number of parameters, theoretically an infinite one, makes very difficult the use of classical control methods and the conventional transducers for position and orientation.” must be moved after the paragraph “An ideal tentacle manipulator is a non-conventional robotic arm with an infinite mobility. It has the capability of takeing sophisticated shapes and of achieving any position and orientation in a 3D space. These systems are also known as hyper redundant manipulators and, over the past several years, there has been a rapid expanding interest in their study and construction. An ideal tentacle manipulator is a non-conventional robotic arm with an infinite mobility. It has the capability of takeing sophisticated shapes and of achieving any position and orientation in a 3D space. These systems are also known as hyper redundant manipulators and, over the past several years, there has been a rapid expanding interest in their study and construction. The control of these systems is very complicated and a great number of researchers tried to offer solutions for this difficult problem. In (Hemami, 1984); (Suzumori et al., 1991) it analyses the control by cables or tendons meant to transmit forces to the elements of the arm in order to closely approximate the arm as a truly continuous backbone. Also, Mochiyama has investigated the problem of controlling the shape of an HDOF rigid-link robot with two- degree-of-freedom joints using spatial curves (Mochiyama & Kobayashi, 1999). Important results were obtained by Chirikjian (Chirikjian, 1993) who laid the foundations for the kinematic theory of hyper redundant robots. His results are based on a “backbone curve” that captures the robot’s macroscopic geometric features. The inverse kinematic problem is reduced to determining the time varying backbone curve behaviour (Takegaki & Arimoto, 1981). New methods for determining “optimal” hyper- redundant manipulator configurations based on a continuous formulation of kinematics are developed. In (Gravagne & Walker, 2001), Gravagne analysed the kinematic model of “hyper-redundant” robots, known as “continuum” robots. Robinson and Davies (Robinson & Davies, 1999) present the “state of art” of continuum robots, outline their areas of application and introduce some control issues. The great number of parameters, theoretically an infinite one, makes very difficult the use of classical control methods and the conventional transducers for position and orientation. The lack of no discrete joints is a serious and difficult issue in the determination of the robot’s shape. A solution for this problem is the vision based control of the robot, kinematics and dynamics. The research group from the Faculty of Automation, Computers and Electronics, University of Craiova, Romania, started working in research field of hyper redundant robots over 25 years ago. The experiments started on a family of TEROB robots which used cables and DC motors. The kinematics and dynamics models, as well as the different control methods developed by the research group were tested on these robots. Starting with 2008, the research group designed a new experimental platform for hyper redundant robots. This new robot is actuated by stepper motors. The rotation of these motors rotates the cables which by correlated screwing and unscrewing of their ends determine their shortening or prolonging, and by consequence, the tentacle curvature (Blessing & Walker, 2004). Segments were cylindrical in the initial prototype, and cone-shaped in actual prototype. The backbone of the tentacle is an elastic cable made out of steel, which sustains the entire structure and allows the bending. Depending on which cable shortens or prolongs, the tentacle bends in different planes, each one making different angles (rotations) respective to the initial coordinate frame attached to the manipulator segment – i.e. allowing the movement in 3D. Due to the mechanical design it can be assumed that the individual cable torsion, Hyper Redundant Manipulators 29 respectively entire manipulator torsion can be neglected. Even if these phenomena would appear, the structure control is not based on the stepper motors angles, but on the information given by the robotic vision system which is able to offer the real spatial positions and orientations of the tentacle segments. Fig. 1. A tentacle arm prototype 2. Kinematics In order to control a hyper-redundant robot it has to develop a method to compute the positions for each one of his segments (Immega & Antonelli, 1995). By consequence, given a desired curvature S*(x, t f ) as sequence of semi circles, identify how to move the structure, to obtain s(x, t) such that * lim ( , ) ( , ) f tt f sxt S xt → = (1) where x is the column vector of the shape description and t f is the final time (see Fig. 2). Fig. 2. The description of the desired shape Y X R 3 R 2 R 1 Z S * Advanced Strategies for Robot Manipulators 30 To describe the tentacle’s shape we will consider two angles ( α, θ ) for each segment, where θ is the rotation angle around Z-axis and α is the rotation angle around the Y-axis (see Fig. 2). To describe the movement we can use the roto-translation matrix considering θ = 2β as shown in Fig. 3. Fig. 3. Curvature and relation between θ and β The generic matrix in 2D that expresses the coordinate of the next segment related to the previous reference system can be written as follow: ( ) ( ) () () ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⋅ ⋅⋅ ⋅ ⋅⋅ − 100 )cos(L 2 cos 2 sin )sin(L 2 sin 2 cos β ββ β ββ (2) In 3D space we cannot write immediately the dependence that exists between two segments. This relation can be obtained through the pre-multiplication of generic roto-translation matrix. One of the possible combinations to express the coordinate of the next segment related to the frame coordinate of the previous segment is the following: :()()()() iiiiiiii generic z y y z RRTrVRR θ αθ =⋅ ⋅⋅ (3) where ( ) ii z R θ and () ii y R α are the fundamental roto-translation matrix having 4x4 elements in 3-D space, and Tr y (V i ) is a 4x4 elements matrix of pure translation in 3-D space and where V i is the vector describing the translation between two segments expressed in coordinate of i-th reference system.e main problem remains to obtain an imposed shape for the tentacle arm. In order to control the robot, we need to obtain the relation between the position of the wires and the position of the segment. Here, a decoupled approach is used for the robot control scheme. Thus the segments are controlled separately, without considering the interaction between them. Considering the segments of the tentacle separately, then ( α, θ ) i is the asigned coordinate of i-th segment. Having as purpose to command the robot to reach the position (α, θ) i the following relation is useful: 0 CB L R θ θ = ∀≠ (4) centre original position of the segment L θ β 2 β [...]... 2 L 12 = CP1 = R + D2 2 L21 = CP3 = R − D1 2 L 22 = CP2 = R − D2 2 (18) where D1 and D2 are the diameters of the segment end discs Based on the Carnot theorem, the lengths A1 and A2 are then obtained: A1 = L2 + L2 − 2 ⋅ L2 ⋅ L2 ⋅ cosθ 11 12 11 12 A2 = L2 + L2 − 2 ⋅ L2 ⋅ L2 ⋅ cosθ 21 22 21 22 (19) The control wires curvature radius R1 and R2 are given by the relations (20 ): R1 = A1 2 ⋅ sin θ 2 R2 = A2... following lengths are computed: L11 = R + D1 2 ⋅ cos(α 1 ) L21 = R + D1 2 ⋅ cos(α 2 ) L31 = R + D1 2 ⋅ cos(α 3 ) L 12 = R + D2 2 ⋅ cos(α 1 ) L 22 = R + D2 2 ⋅ cos(α 2 ) L 22 = R + D2 2 ⋅ cos(α 3 ) (22 ) 35 Hyper Redundant Manipulators where αn is according to Fig 5: ⎧α 1 ⎪ ⎨α 2 ⎪α ⎩ 3 −α = = 120 ° − α = 24 0° − α (23 ) Based on (19) and (20 ) the curvature radiuses R1, R2 and R3 of the three control wires are then... ⎢Δy ⎥ = α 2 ⋅ 2 − (D2 + x ) ⎣Δy ⎦ ⎣ s2 ⎦ (76) 44 Advanced Strategies for Robot Manipulators and the orientation angles for each plane will be tgθ s = Δx s1 Δy s1 = Δx = tgθ Δy (77) hence θ s ( s' ) = θ ( s ) , s ∈ [ 0 , l ] , s'∈ [ 0 , l' ] for the plane ZS1 OS1 YS1 and tgq s = X s1 Δz s 2 Δy s 2 = Δz Δy Zs 2 Cs1 Fθs x s1 O s1 d Cs1 y s1 (78) zs 2 Ys1 d Cs 2 Cs Fq s Os 2 2 qs Ps 2 ys 2 Ys 2 Fig 9 Image... ⎤ ⎡ o11 ⎤ ⎪ ⎡c x1 ⎤ ⎨⎢ ⎥ − ⎢ ⎥ ⎬ + ⎢c ⎥ ⎪⎣ y ⎦ ⎣o 12 ⎦ ⎪ ⎢ y 1 ⎥ ⎭ ⎣ ⎦ ⎩ (73) ⎫ ⎧ ⎪⎡ z ⎤ ⎡ o21 ⎤ ⎪ ⎡c z 2 ⎤ ⎨⎢ ⎥ − ⎢ ⎥ ⎬ + ⎢c ⎥ ⎪⎣ y ⎦ ⎣o 22 ⎦ ⎪ ⎢ y 2 ⎥ ⎭ ⎣ ⎦ ⎩ (74) for the ZS1 OS1 YS1 frame and ⎡ zs 2 ⎤ 2 ⋅ R(φ ) ⋅ ⎢y ⎥ = α 2 ⋅ 2 − (D2 + x ) ⎣ s2 ⎦ for the ZS2 OS2 YS2 frame, where [ c x1 , c y1 ]T and [ c z2 , c y 2 ]T the image centers, α 1 and α 2 are the scale factors of the length units in the... 22 21 22 (19) The control wires curvature radius R1 and R2 are given by the relations (20 ): R1 = A1 2 ⋅ sin θ 2 R2 = A2 2 ⋅ sin θ 2 (20 ) Finally, the lengths of the control wires are obtained as in (21 ): Lw 1 = R1 ⋅ θ = A1 ⋅ θ / 2 ⋅ sin θ 2 Lw 2 = R2 ⋅ θ = A2 ⋅ θ / 2 ⋅ sin θ 2 (21 ) For the 3D case, a virtual wire is considered, which gives the α direction of the curvature Considering one virtual wire... = Lw 2 + ΔR ⋅ θ ⋅ cos(α 2 ) ( 12) 33 Hyper Redundant Manipulators Replacing the (8) we obtain θ in function of α: θ= 2 Lw 1 − Lw 2 ⋅ ΔR 3cos (α ) − 3 sin (α ) (13) And considering the eq (10) for the third wire: Lw 3 = Lw 1 + ( 2 ⋅ ( Lw 1 − Lw 2 ) ⋅ 3cos (α ) − 3 sin (α ) 3 cos (α ) − 3 sin (α ) ) (14) Finally the α angle can be obtained using the function atan2 α = atan2 ( 3 ( Lw 2 − Lw 3 ) , 2 Lw... si ( si ) ≠ π 2 , we obtain Δθ ( Δs') = i=1 1 ( Δxs1 ( Δs') cosθs1 ( Δs')) Δs' i =2 Δθ ( 2 ⋅ Δs') = ( Δx1 ( 2 ⋅ Δs') / Δs'− cos θ s1 ( Δs') ⋅ Δθ ( Δs') ) cosθ s1 ( 2 ⋅ Δs') i=m ⎛ m −1 ⎝ j =1 Δθ (m ⋅ Δs') = ⎜ Δx1 (m ⋅ Δs') / Δs'− − ⎜ ∑ cos θ s1 ( j ⋅ Δs') ⋅ Δθ ( j ⋅ Δs') ⎞ 1 ⎟⋅ ⎟ cos( m ⋅ Δs' ) ⎠ (100) 48 Advanced Strategies for Robot Manipulators If θ si (si ) = π 2 , a similar procedure for y s1 (i ⋅... [ ] Fu = Fu (t ), t ∈ 0 , t f 42 Advanced Strategies for Robot Manipulators From (60)-( 62) , it results, s s ρ ∫ ∫ (q′(sin q′ sin q′′ cos(q′ − q′′) + cos q′ cos q′′) − θ ′ cos q′ sin q′′ sin(θ ′′ − θ ′) + 00 + q 2 (cos q′ sin q′′ cos(θ ′ − θ ′′) − sin q′ cos q′′) + θ 2 cos q′ sin q′′ cos(θ ′ − θ ′′) − q′q′′ sin(q′′ − q′))ds′ds′′ + s + ρg ∫ cos q′ds′ + 0 1 2 kd q = Fq 2 s s ρ ∫ ∫ (q′ sin q′ cos q′′... the length of each wire For the hard wire, made from the same material as the central bone, and by consequence having the same elasticity, referring to Fig 4, we can write: X1 Hard wires Z1 Lw3 Lw2 Lw1 X Soft wires R2,R3 R1 θ Y X2 Z Y1 Z2 Y2 Fig 4 Different types of wires ⎧ Lw 1 ⎪ ⎨Lw 2 ⎪L ⎩ w3 = R1 ⋅ θ = R2 ⋅ θ = R3 ⋅ θ (5) For the soft wires, we can write: ⎧ ⎪L w1 ⎪ ⎪ ⎨Lw 2 ⎪ ⎪L ⎪ w3 ⎩ = [R1 ⋅θ ]⋅... sin(θ / i ) = [R2 ⋅ θ ]⋅ θ /i sin(θ / i ) = [R3 ⋅ θ ]⋅ θ /i (6) where Lwn is the length of the n-th wire and Ri is the radius of the curvature of the real i-th wire Farther it can be written: Rn = ( R − ΔR ) ⋅ cos(α n ) where ΔR is constant equal to the distance between the center and the wires and αn is: (7) 32 Advanced Strategies for Robot Manipulators ⎧α 1 ⎪ ⎨α 2 ⎪ ⎩α 3 = = = −α 120 ° − α 24 0° − α (8) . A 2 are then obtained: 22 22 111 12 11 12 22 22 221 22 2 122 2cos 2cos ALL LL ALL LL θ θ =+−⋅⋅⋅ =+−⋅⋅⋅ (19) The control wires curvature radius R 1 and R 2 are given by the relations (20 ):. 11 1 1 12 2 1 21 1 2 22 2 2 31 1 3 22 2 3 2cos( ) 2cos( ) 2cos( ) 2cos( ) 2 cos( ) 2 cos( ) LRD LRD LRD LRD LRD LRD α α α α α α =+ ⋅ =+ ⋅ =+ ⋅ =+ ⋅ =+ ⋅ =+ ⋅ (22 ) Hyper Redundant Manipulators. 11 22 2sin 2 2sin 2RA RA θ θ = ⋅=⋅ (20 ) Finally, the lengths of the control wires are obtained as in (21 ): 11 1 22 2 2 2 /2 sin /2 sin w w LR A LR A θ θ θθ θθ =⋅=⋅ ⋅ =⋅=⋅ ⋅ (21 ) For the