Optimum Static Balancing of the Parallel Robot for Medical - Download as PDF

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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.