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12th IFToMM World Congress, Besançon (France), June18-21, 2007 Optimum Static Balancing of the Parallel Robot for Medical 3D-Ultrasound Imaging S. Lessard*, P. Bigras, I. A. Bonev S. Briot, V. Arakelian Département de génie de la production Département de Génie Mécanique et Automatique, automatisée L.G.C.G.M. – EA3913 École de Technologie Supérieure (ETS) Institut National des Sciences Appliquées (I.N.S.A.) 1100 Notre-Dame St. W, Montréal, QC 20 Buttes de Coësmes Av. – CS 14315 Canada, H3C 1K3 F-35043 Rennes, France Abstract - Static balancing of mechanical systems is useful (Fig. 2). A robot suited for the quantification of the lower and required in many situations. The objective of such balancing limbs arteries could perform any less restricting arterial is the compensation of gravitational forces in order to achieve a examination since it is the longest arterial examination. static equilibrium. A balanced system becomes safer and actuators are reduced in size. However, balancing a system The linear displacement was chosen long enough to cover requires numerous complex mechanical add-ons or unavoidable the whole body from neck down, thus preventing addition of mass. This is the reason why methods of partial static repositioning of the patient for different examinations or balancing have been developed and applied in practice. In this having to place him meticulously in the robot workspace. paper, a newly designed parallel robot for medical 3D- Designing a robot for a precise confined workspace would ultrasound imaging is required to be statically balanced without have led to a less practical system. complicated design modifications. Simple mechanical add-on that is optimally designed can reduce substantially the effect of gravity. The efficiency of these suggested solutions is illustrated by numerical simulation of the robot. Keywords: static equilibrium, parallel robot, medical robot, root-mean-square minimization I. Introduction Recently, parallel robotics has broken through new field of research such as high speed manipulation, material handling, motion platforms and medical equipment. In the latter, a new robot has been developed at École de technologie supérieure (Fig. 1). This robot is designed to perform an ultrasound scan of a human patient’s arteries. It consists of two five-bar planar mechanisms, which are connected to a tool holder by an articulated telescopic Fig. 1. CAD model of the prototype parallel robot strut. Four motors are mounted inside the main frame while a fifth motor is attached directly to the tool holder Axis Dimension (mm) Description allowing large amplitude tool rotations. The whole frame X 1500 Span (arteries axis) is mounted on a linear motor which allows large Y 500 Lateral horizontal displacements of the robot. Thus, such an Z 500 Vertical architecture has six degrees of freedom and ensures the TABLE I. Positioning workspace dimensions required geometric and kinematic characteristics (see Axis Orientation (degree) Description tables I and II) [1]. Around X -75° to 90° Span (arteries axis) The robot workspace was defined considering the Around Y -50° to 35° Lateral examination of one half of the human body, cut vertically Around Z -45° to 45° Vertical TABLE II. Workspace orientations * E-mail: simon.lessard.3@ens.etsmtl.ca A partnership with Institut National des Sciences Appliquées (Rennes, France) has led to this present work on optimum static balancing of the six-degree-of-freedom medical robot. 12th IFToMM World Congress, Besançon (France), June18-21, 2007 III. Complete Static Balance Counterweights mounted on each moving links achieve complete static equilibrium. Fig. 3 illustrates the complete static balance of the parallel robot for medical 3D- ultrasound. Fig. 2. Required workspace for the robotized ultrasound examination The first virtual prototype version of the robot (Fig. 1) is smaller resulting in a smaller workspace. For assembly considerations, symmetric motors of each five-bar mechanism are not mounted vis-à-vis. The robot design next step is the minimization of the input torques. This goal is well-known in the field of robotics and is composed of the static balancing of Fig. 3. Complete balancing by counterweights moving masses. Static balancing is typically achieved by adding a Clearly, the added counterweights increase the robot’s counterweight or a spring to each bar. Complete static inertia. Also, the tool holder motor would be difficult to balancing is carried out by keeping constant the potential assemble with a counterweight attached because the joint energy of the system for all configurations. In previous is not a ball joint, but a combination of a universal joint studies, a number of methods have been proposed for and a revolute joint. gravity balancing of robotic systems: spring balancing of a In the case of complete balancing with springs, it is planar pantograph [2, 3], n-springs solutions to a single necessary to add auxiliary mechanisms (parallelograms), attached point [4], balancing of a spatial positioning table which lead to a very complicated mechanical architecture. [5] and 6-dof parallel manipulators [6, 7], as well as A mechanical system that is completely balanced several auxiliary mechanisms for spring support [8-12]. becomes either very heavy or too complicated to This article deals with the optimum static balancing of a manufacture and assemble. That is why we developed a parallel robot developed for medical 3D-ultrasound partial balancing approach. It is obvious that the load imaging. The suggested analytical solution allows a reduction on actuators is partial, but the design solution is significant reduction in the input torques by means of very simple. simpler design solutions. The paper is organized as follows. Firstly, the input torques due to the static and IV. Input Torques dynamic loads are examined and a simplified calculation The input torque of the ith actuator can be expressed as: approach is proposed. Then an optimal static balancing ∑ (J ) ⋅ G 13 method is developed, which is formulated by the input τi = T j i j (i = 1,..., 6) (1) torques root-mean-square values minimization. j =1 II. Static and Dynamic Models where (JTj)i corresponds to the ith line of the transposed Jacobian matrix between the center of masses Sj of the In our medical robotic application, motion is slow and link j and the actuated variables qi. Gj is the gravity forces smooth. Therefore, the end-effector velocity and of the link j (Fig. 4-6). acceleration are small. Our observations show that the The structure of this robot has some particularities input torques caused by the inertia effect of the moving which allow a considerable simplification of torques masses are very small compared to the input torques determination. The tool holder represents an Assur group caused by the gravity force. Thus, inertia effect can be [13] with two links and three joints (spherical, universal neglected. For example, the variation of the input torques and revolute pairs) dividing the robot architecture in two for a prescribed trajectory (with 0.6m/s2 maximal parts. Thus the tool holder can be disconnected from the acceleration) shows that the maximal difference between robot structure to be examined separately. the dynamic and the static loads is less than 1%. This means that the motor torques are mainly caused by gravity. 12th IFToMM World Congress, Besançon (France), June18-21, 2007 where mi, and ri are respectively the mass and position of roll the centre of masses of the ith link, g is the gravity and m6 z γ is the mass of actuator 6. H The horizontal reaction forces are projected from axis w 12 to axis y: RHy = (m12 g − RHz ) tan β sin α (5) pitch 11 β REy = − RHy (6) y E x 13 α yaw w Fig. 4. Euler angles of the tool From this point forward, the tool is considered statically balanced around the sixth motor, i.e. the tool’s center of mass coincides with the motor rotation axis. Such an approach simplifies reaction forces and input torques calculations. The reaction forces RE and RH (respectively applied by the tool holder at points E and H) are function of the yaw Fig. 6. Schematics of the two five-bars mechanisms and pitch angles only, the roll angle being equal to the with applied reaction forces sixth motor rotation angle, this motor input torque being independent of the other motors. Thus, it is possible to deduce the potential energy V of the first five bar mechanism (Fig. 6): V = ( m2 z 2 + m3 z 3 + m7 z 7 + m8 z8 ) g + REz z E + REy y E (7) r12 L with z 2 = z A + r2 cos q 2 (8) H r13 z 3 = z C + r3 cos q3 (9) β r11 r7 z 7 = ( L2 cos q2 + z A − z E ) + zE (10) G12 L7 z r8 G13 E z8 = ( L3 cos q3 + z C − z E ) + zE (11) G11 L8 w where zi is the vertical coordinate of the centre of mass Si Fig. 5. Schematic of the tool assembly of the ith link and r2=AS2, r3=CS3, r7=ES7, r8= ES8. Then the input torques are determined by differentiating The length L of the passive linear joint is calculated as a the potential energy by the motor articulation position function of the orientation angles: vector Q. To transform the reaction forces at the tip into reaction motor torques, a Jacobian matrix is needed [14]: cos β L = Lf (2) cos α ⎡τ ⎤ ∂V ∂V τ = ⎢ 2⎥ = − JT (12) where L f is the distance between the two planar five bar ⎣τ 3 ⎦ ∂Q ∂E mechanisms: L f = x E − x H . where τi is the ith actuator torque, Q = [q 2 , q3 ] and J is the Thus, the vertical reaction forces on point H and E can five-bar mechanism’s Jacobian matrix [15]. The obtained be determined from the static equilibrium equations: expression of τ is given in appendix 1. ( m12 ( L − r12 ) + m11r11 + m13 r13 ) g The second five-bar mechanism’s torques τ4 and τ5 of RHz = (3) actuators 4 and 5 are determined in a similar way. L REz = ( m6 + m11 + m12 + m13 ) g − RHz (4) 12th IFToMM World Congress, Besançon (France), June18-21, 2007 V. Input Torques Root-Mean-Square Values ∂f i ∂f i Minimization = 0 and =0 (17) ∂Ci ∂q ki The input torques minimization is carried out by each from which the coefficients are solved: actuator torque root-mean-square values minimization for all the workspace. The workspace used in the calculation q ki = 2 arctan(t mi ) (m=1,…,6) (18) is only the yz plane because the linear motorized axis is not subject to gravity balancing. Three solutions are −∑τ sin(q − q ) WS i i ki Ci = (19) considered for optimum balancing. ∑ sin (q − q ) WS 2 i ki A. Tension / Compression Spring Equilibrium where tmi are the roots of a polynomial pi(t) given in Firstly, one solution is an equilibrium mechanical appendix 2. system composed of zero free length springs (tension or compression) attached on each motorized arms. The fixed B. Torsion Spring Equilibrium end of each spring is positioned optimally to release the Secondly, static balance can be achieved using torsion actuator. spring mounted on each actuator axes. The spring torque is linear to the bending angle: τ ki = K (qki − qi ) (20) Fig. 7. Spring balancing The spring applied torque’s potential energy is: K Vki = i [(z ki cos q ki − lki cos q i ) 2 + 2 (13) (z ki sin q ki − lki sin q i ) 2 ] Fig. 8. Torsion spring balancing where Ki is the stiffness spring coefficient, which is attached on a fixed arm of length zki and orientation qki The minimization conditions are the same: (Fig.7) and qi is the angular position of the ith actuator. ∑ (τ 2 The other end is linked on the motorized arm at length lki. fi = + τ ki ) → min ( i=2,…,5) (21) i By differentiating equation (13), we determine the input WS Ki ,qki torque: ∂f i ∂f i ∂V = 0 and =0 (22) τ ki = ki = K i l ki z ki sin(qi − qki ) (14) ∂K i ∂q ki ∂qi The spring stiffness coefficient Ki and attached linear from which the coefficients are solved: positions zki and lki are consolidated into parameter Ci. Thus, the optimal calculated Ci will position precisely a ∑τ q ∑ q − ∑τ ∑ q WS i i WS i WS i WS 2 i certain spring Ki on the arms: qki = (23) N ∑τ q − ∑τ ∑ q i i i i τ ki = Ci sin(qi − qki ) (15) WS WS WS Two parameters have to be optimized for each motor: ∑τ (q − q )WS i ki i the constant Ci and the angular position of the fixed arm K =− (24) ∑ (q − q ) i 2 qki. ki i WS The torque root-mean-square value minimization leads to the following condition: where N is the number of calculated positions in the ∑ (τ workspace. 2 fi = i + τ ki ) → min (i=2,…,5) (16) Ci ,qki WS C. Counterweight Equilibrium where WS := Workspace. For this purpose, these conditions must be satisfied: Lastly, a balancing approach is carried out by adding a counterweight of mass Mi on each motorized axis. 12th IFToMM World Congress, Besançon (France), June18-21, 2007 The static torque root-mean-square sum minimization was reduced up to 50% ÷ 91.5%. In more practical terms, the maximum motor torque required was reduced up to 34.3% ÷ 46.7% . Fig. 10 shows input torques variations for unbalanced and optimum balanced robots with extension springs (the simulation was carried out in a static mode or operation). Fig. 9. Counterweight balancing The mass Mi is placed at a certain angle so that the maximum torque applied is at a specific desired motor angle. The optimum counterweight system is then function of the mass Mi and position qci for a given length lci. τ ci = − M i lci sin(qi − qci ) (25) The mass Mi and length lci are consolidated into a single constant Ci: (a) Actuator 2 τ ci = −Ci sin(qi − qci ) (26) Determination of parameters Ci and qci is equivalent to the determination of parameters Ci and qki in section A. VI. Results The proposed robot (Fig. 1) with above mentioned geometrical parameters and mass distribution (appendix 3) was used for numerical simulation. Three mechanical solutions were tested. As a mean of comparison, reductions of the RMS and maximum motor torques are (b) Actuator 3 given in tables (III)-(V). Tension/Compression spring Max Motor RMS Parameters Torque 2 qk2 = 1.7o C2 = 0.104 Nm 57.6% 34.7% 3 qk3 = -42.7o C3 = 0.263 Nm 91.2% 45.6% 4 qk4 = 3.4o C4 = 0.081 Nm 50.1% 33.1% 5 qk5 = -43.0o C5 = 0.187 Nm 90.2% 43.1% TABLE III. Tension / compression spring optimum configurations Max Motor Torsion spring Parameters RMS Torque (c) Actuator 4 2 qk2 = 7.7o K2 = -4.55 Nm/rad 56.9% 45.4% 3 qk3 = 180o K3 = 5.29 Nm/rad 90.4% 45.2% 4 qk4 = 6.2o K4 = -3.33 Nm/rad 50.0% 46.7% 5 qk5 = 180o K5 = 3.74 Nm/rad 89.2% 34.3% TABLE IV. Torsion spring optimum configurations Max Motor Counterweight Parameters RMS Torque 2 qc2 = -178.4o C2 = 4.63 Nm 57.6% 34.7% 3 qc3 = -221.4o C3 = 11.63 Nm 91.5% 46.4% 4 qc4 = -176.9o C4 = 3.62 Nm 50.1% 33.1% 5 qc5 = -224.2o C5 = 8.36 Nm 90.2% 44.1% (d) Actuator 5 TABLE V. Counterweight optimum configurations Fig. 10. Variation of the actuator torques for the orientation angles α=β=γ=0 deg 12th IFToMM World Congress, Besançon (France), June18-21, 2007 VII. Conclusion Appendix 1 The perfect static balancing of a spatial multibody τ 2 = − M 1 sin q 2 + 2 L2 (− y E cos q 2 + ( z E − z A ) sin q2 ) ⋅ mechanical system can eliminate completely the load (2M 3 ( z E − zC − L3 cos q3 ) + 2M 4 (− y E + L3 sin q3 )) / A caused by the gravity force. However, such a solution τ 3 = − M 2 sin q3 + 2 L3 (− y E cos q3 + ( z E − zC ) sin q3 ) ⋅ leads to inevitably complicated design add-ons or to unavoidable increase in total mass. In most cases, the (2M 3 (− z E + z A + L2 cos q2 ) + 2M 4 ( y E − L2 sin q2 )) / A complete balance is achieved only in theory but not in where practice. In this paper, a parallel robot for medical 3D- A = 4 ( L3 (− y E cos q3 + ( z E − z A ) sin q3 ) − ultrasound imaging was optimally statically balanced. The L2 (− y E cos q2 + ( z E − zC ) sin q2 ) + L2 L3 ⋅ proposed simple solutions presented a partial balancing sin(q2 − q3 ) + y E ( z A − zC )) effect minimizing the actuator torques. Future work will investigate more efficient minimization M 1 = g ( m2 r2 + m7 L2 r7 / L7 ) by increasing the number of variables to the problem. The M 2 = g (m3 r3 + m8 L3 r8 / L8 ) development of a prototype with the suggested balancing M 3 = g (m7 ( L7 − r7 ) / L7 + m8 ( L8 − r8 ) / L8 + REz ) system is planned. M 4 = REy References Appendix 2 [1] S. Lessard, I. Bonev, P. Bigras, L.-G. Durand, G. Soulez, G. Cloutier, and J. A. DeGuise, "Parallel Robot for Medical 3D- pi (t ) = a6 t 6 + a5 t 5 + a 4 t 4 + a3 t 3 + a 2 t 2 + a1 t + a0 Ultrasound Imaging," presented at International Symposium on Industrial Electronics, École de technologie supérieure, Montréal, with Canada, 2006. a0 = b5 , a1 = 2b7 , a2 = −3 b5 + 4 b8 , a3 = −4b7 + 8b6 , [2] J. L. Herder, "Energy-Free Systems. Theory, conception and design of statically balanced mechanisms", PhD Thesis: Delf University of a4 = −a 2 , a5 = a1 and a6 = −a0 Technology, 2001. where ∑ ∑ ∑ sin [3] E. Shin and D. A. Streit, "Spring equilibrator theory for static 2 balancing of planar pantograph linkages," Mechanism & Machine b0 = − τ i sin qi , b1 = τ i cos qi , b2 = ( qi ) , Theory, vol. 26, pp. 645-657, 1991. WS WS WS = ∑ cos = −2∑ cos q sin q [4] G. J. Walsh, D. A. Streit, and B. J. Gilmore, "Spatial spring 2 equilibrator theory," Mechanism & Machine Theory, vol. 26, pp. b3 (qi ) , b4 i i 155-170, 1991. WS WS [5] J. Wang and C. M. Gosselin, "Static balancing of spatial three- degree-of-freedom parallel mechanisms," Mechanism & Machine b5 = ∑ (τ b cos q + b cos q sin q ) , WS i 2 i 0 i i Theory, vol. 34, pp. 437-452, 1999. [6] I. Ebert-Uphoff, C. M. Gosselin, and T. Laliberte, "Static balancing of spatial parallel platform mechanisms - revisited," Journal of b6 = ∑ (τ b sin q − b cos q sin q ) , i 3 i 1 i i Mechanical Design, Trans. of the ASME, vol. 122, pp. 43-51, 2000. WS [7] C. M. Gosselin and J. Wang, "Static balancing of spatial six-degree- of-freedom parallel mechanisms with revolute actuators," Journal of b7 = ∑ (τ WS i (b4 cos qi + b2 sin qi ) + b0 (sin 2 (qi ) Robotic Systems, vol. 17, pp. 159-170, 2000. [8] A. Fattah and S. K. Agrawal, "On the design of a passive orthosis to gravity balance human legs," Journal of Mechanical Design, − cos 2 ( qi )) + b1 cos qi sin qi ) Transactions of the ASME, vol. 127, pp. 802-808, 2005. [9] A. Fattah and S. K. Agrawal, "Gravity-Balancing of Classes of b8 = ∑ (τ WS i (b3 cos qi + b4 sin qi ) + b10 (sin 2 ( qi ) Industrial Robots," presented at International Conference on Robotics and Automatics, Orlando, Florida, 2006. − cos 2 ( qi )) − b0 cos qi sin qi ) [10] I. Simionescu and L. Ciupitu, "Static balancing of the industrial robot arms. Part I: discrete balancing," Mechanism and Machine Theory, vol. 35, pp. 1287-1298, 2000. Appendix 3 [11] I. Simionescu and L. Ciupitu, "Static balancing of the industrial g = 9.81 m/s², m2 = m4 = 1.235kg, m3 = m5 = 1.549kg, robot arms. Part II: continuous balancing," Mechanism and Machine m6 = 0.331kg, m7 = m8 = m9 = m10 = 0.536kg, Theory, vol. 35, pp. 1299-1311, 2000. [12] J. Wang and C. M. Gosselin, "Passive mechanisms with multiple m11 = 0.107kg, m12 = 0.083kg, m13 = 0.111 kg, equilibrium configurations," Transactions of the Canadian Society L2 = L4 = 0.5m, L3 = L5 = 0.7m, L7 = L8 = L9 = L10 = 0.6m, for Mechanical Engineering, vol. 28, pp. 139-151, 2004. Lf=0.1m, zA = zF = 0.13m, zC=zK = 0.28m, r2=r4= 0.2094m, [13] V. Zinoviev, Théorie des mécanismes et des machines, Ed. de la r3 = r5 = 0.3046m, r7 = r8 = r9 = r10 = 0.3m, r11 = 0.0355m Paix (Moscou), 215p., 1975. [14] R. Clavel and K. Miller, "The Lagrange-Based Model of Delta-4 and r12 = 0.0315m. Robot Dynamics," Robotersysteme, vol. 8, pp. 49-54, 1992. [15] C. Gosselin and J. Angeles, "Singularity analysis of closed-loop kinematic chains," IEEE Transactions on Robotics and Automation, vol. 6, pp. 281-290, 1990.

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