Inverse kinematic deriving and actuator control of Delta robot using symbolic computation technology

Feng Lihang Zhang Weigong， Lin Guoyu Gong Zongyang Chen Gang

(1School of Instrument Science and Engineering， Southeast University， Nanjing 210096， China)

(2Suzhou Research Institute， Southeast University， Suzhou 215000， China)

(3School of Mechanical Engineering， Nanjing University of Science and Technology， Nanjing 210094， China)

T he robot Delta，which was initially developed by Clavel in 1985，is a famous spatial parallel mechanism allowing three translational degrees offreedom(DoF)［1］.Due to the superior qualities of large workspace， high speed and weak kinematics coupling， Delta is drawing more and more attention of scholars and engineers.

To model this mechanism，the proposed ways of kinematic solving are mainly covered by the analytical method and the numerical approach［2］.Earlier studies focused on the analytical method and closed-form solutions.Kinematic singularity and optimal design are discussed a lot by Clavel et al［3-5］.But these can be cumbersome with hand derivations.Followed by the numerical method， which invokes the iterative solver of nonlinear equations with mathematical engines，researchers need to better understand the mechanism in advance so that constraint equations can be programmed and solved.Thus so far， the widespread system-level solving procedure is always implemented on several steps such as the established physical model in Pro/E， Solidworks， etc.， the kinematic analysis in ADAMS，and numerical iteration in Matlab with every time step， etc.However， complexity and low computational efficiency exist in the procedure，and the numerical expression does not give a distinct symbolic representation.Recently， the symbolic technology of the graph theory has been applied to a mechanical system.Formulating symbolic equations attracts much interest due to the advantages of integrative modeling，automatic removal of multiplications and trigonometric simplifications， etc［6］.McPhee et al.［7］further developed an approach that the mechanism's topology was modeled with a linear graph.Also， several examples such as slide-crank mechanism，a planar 3-DoFs robot and a general openloop robot have been implemented［8］.Though researchers claimed that symbolic computation can be applied to more complex robots with closed-loops，few cases have been reported to date， especially on Delta.Since a general symbolic computation engine such as Maple，MuPad and Mathematica is required;that is，they can be coded into routines and run while simulation codes are being processed without providing users to manipulate the underlying equations.We apply similar applications on Delta.

In this paper，the multibody analysis of Delta on coordinate selection and how to manipulate the symbolic equations are given.Explicit symbolic expression of constraints and inverse kinematic solutions are obtained by using a computation engine—MapleTM.Finally， actuator control can be directly realized，and the correctness and precision are verified with trajectory tracking.

1 Principle of Symbolic Computation on Delta Robot

1.1 Linear graph theory applied to mechanical systems

In the linear graph of a mechanical system，different spanning trees in conjunction with many algorithms have been developed to describe their topology，which is proved to be a convenient method［9］. Definitions of nodes， edge， circuit， tree and subgraph for a mechanical system have been described as well.Rigid body elementsm，which start at the ground node，end at the node representing a reference frame at the center of mass.Rigid arm elementsr，which are used to define new reference frames relative to the mass center，start at the mass center and end at the desired node.Joint elementsj， which define the allowable motions between two bodies comprising a kinematic pair，contains different edge types for different joints such as revolute jointsh， prismatic jointss，universal jointsu， ball jointsband translational jointst.After all elements are defined，physical modeling can be established.Since we focus on the kinematics， the system dynamics is beyond the current scope and the procedure is simply processed as follows:

1)Linear graph representation.The fundamental circuit subsets，which provide closure conditions around any loops and are satisfied with the associated edge across(translation， rotation)variables， are primarily taken into account.

2)Spanning tree selection for coordinates.When a tree is selected for the graph，the circuit equations can be used to express all the kinematic variables，and the branch coordinates q are defined.

3)Constraint equations projection and simplification.The constraints are generated by projecting the circuit equations for cotree joints onto the reaction space.The closed chain with an incidence matrix representation of the linear graph agrees well with the dependent branch coordinates［10］.Thus， one obtainsmnonlinear algebraic equations in terms of thenbranch coordinates q:

The system's DoF can be given byf=n－m.

As an example，the slide-crank mechanism is depicted in Fig.1.Rigid arm elementsr1tor4are selected for the tree of the graph with kinematic transformations since no unknown coordinates or variables are introduced into q.By selecting theh1，h2，s1into the tree， the joint coordinate is set as

where β is the revolute joint angle andsis the prismatic displacement.Then the reaction space forh2is spanned by unit vector i and j(the directions of the joint reaction forces)，onto which the circuit equation forh2is projected:

where p can be i or j，and Riis the translational vector of element.Substituting the elemental constitutive equations，for instance，=0，and evaluating，we obtain whereL12andL34are the length of the two arms，respectively.Thus，we obtainm=2 constraint equations in terms of then=3 branch coordinates for this 1-DoF system.This have been demonstrated by McPhee［9］and one can use symbolic computing to time-differentiate the position-level constraint Eq.(1).

Fig.1 Linear representation example.(a)Slider-crank mechanism;(b)Linear graph of slider-crank

1.2 Symbolic representation of Delta robot

Similarly，we apply the above procedure to the Delta robot which consists of a moving platform connected to a fixed base through three parallel chains with 120°away from each other(see Fig.2 and Fig.3).Each chain contains a revolute joint activated by an actuator on the base.Movements are transmitted to the moving base through parallelograms formed by bars and spherical joints.Especially，a couple of spherical joints in each leg can be replaced by universal joints because the parallelogram structure makes an extra constraint for the 3-DoF translational motion［4］.

Since the Delta has a complete symmetrical topology，the symbolic representation is determined only by choosing one chain.Just like the virtual jointvh12depicted in Fig.1(b)，we use a jointt0which allows only three translational DOFs for Delta，and then it can translate the full linear graph into a subgraph with a single chain(see Fig.3(b)，dot line).In this subgraph，spherical jointsb11-b12are chosen while they are excluded from all single or separate trees because there are no variables appearing in equations.As a result，joint coordinates with a set of constraints are reduced，and the single spanning tree will include the following elements:rigid bodies(r10-r17，r'13-r'16)，revolute jointh11，universal jointsu11-u12and virtual jointt0.Each revolute joint contributes 1 coordinate，the universal joint contributes 2 and the virtual joint contributes 3，so a total of 8 joint coordinates can be obtained as

Fig.2 Delta mechanism with vector coordinates

Fig.3 Delta representation.(a)One typical kinematic chain;(b)Linear graph representation

Note that the universal joint can be dissociated into two orthogonal revolute joints，and the parallelogram structure makesu11a=u12aandu11b=u12b.The resulting set of coordinates is reduced to 6 with

The constrains associated with legk(k=1，2，3)can be acquired by projecting the circuit equations onto the reaction space forb11andb12.By substituting variables，the constraints are of the general form as

where θkis the driving angle of jointh11;αkand βkrefer to universal joint angles ofU11aandU11b;andr(t)refers

to the prescribed motionxt0，yt0andzt0.Giving an insight into Eq.(7)with the joint dissociation of αkand βk，it is simplified as

which indicates that inverse solutions of Delta can be obtained by only solving one single kinematic chain.The velocity and acceleration equations can be obtained by taking the derivative of Eq.(8)with respect to time.Apparently，the general form Eq.(8)is a little different from conventional vector loops solutions［11］which are in the form of three driving angles θkand three translation positionsr(t)，but in fact，results will be the same when solving.

2 Simulation and Symbolic Verification

2.1 Physical model and multibody analysis

To confirm the symbolic representation，the physical model of Delta is built in MapleSim［10］so that mechanical components can be defined based on the linear graph.Fig.4 depicts the model and the parameters are given as follows:the revolute joint is 0.25 m away from the fixed frame with an orientation angle of－π/6;the driving arm is 0.4 m in length;two sides of the parallelogram are 0.1 m and 1 m in length，respectively;the moving base has a radius of 0.05 m.Parameters are chosen generally for easy computation so that the Pythagorean theorem is satisfied in chains when all the driving angles equal 0.In this case，the calculated workspace is defined by the driving angle as θi∈(－ arccos(1/3)，π － arccos(1/7)］andz＜ 0.Thus，the geometrical singular is avoided when solving inverse kinematic.Fig.4(b)depicts the range of arm motion for better understanding.

Fig.4 Delta robot model(unit:m).(a)3D physical model;(b)Geometrical singularity

2.2 Symbolic manipulation of inverse kinematic

When formulating a mechanical system's equations，there are some coordinate selections in the optimization techniques.The optimization procedure always requires multiple evaluations of objective functions and might be very tedious.By using the indirect joint coordinate［8］，we obtain a result that 18 variables are given in a set of 15 constraints equations(3 for moving base motion ofX，YandZ，1 for revolute joint in driving arm，4 for a couple of universal joints).A snapshot of the 5 constraint equations set is shown in Fig.5，wherex(t)，y(t)，z(t)correspond to the desired motion of moving base，parameterArepresents the orientation angle，and the variables α(t)，β(t)and θ(t)are universal joint angles and revolute joint，respectively.Obviously，this inverse symbolic representation is in the form of Eq.(8).Note that the latter four constraints have duplications due to the universal jointu11andu12.By only solving the former three equations for θ(t)，the explicit symbolic representation of the kinematic solution can be easily obtained(see Fig.5).

3 Actuator Control and Trajectory Tracking

To verify the symbolic solution，the simulation of block components are created by using the derived equations so that an controller is designed.Here，the controller can be made for each single chain with three input variables of desired motionX，YandZ，one output variable of driving angle θ(t)and one orientation angle.By using a virtual electrical driving subsystem(see Fig.6)，the kinematic relations for any desired trajectory can be evaluated.

Fig.6 Subsystem diagram with PID control for a single kinematic chain

Two motion curves are chosen for trajectory tracking，respectively.One is the circular path in theX-Yplane used for the correctness test under ideal conditions(see Fig.7);the other is Adept motion［12］which is always used as a benchmark test in Pick-and-place operation(see Fig.8).

Fig.7 Circular path for kinematic response.(a)Circular path in X-Y plane;(b)Driving angles of inverse kinematic

The desired circular trajectory(see Fig.7)，in which a straight line is inserted at the start of the path to test whether trajectory change will have an effect on the results or not，is compared against the actual point track.Note that both the straight line and circular segments agree with the kinematic motion well.For Adept motion(see Fig.8)，the inverse kinematic solution is verified by a PID controller with position feedback.The trajectory，point track，driving angles and errors can be observed as well.As expected，the simulation satisfies the requirement of the motion and the trajectory error is within acceptable thresholds for kinematic response.

4 Conclusion

According to the linear graph representation of the Delta robot，the inverse kinematic can be derived with a symbolic form.The symbolic equations representation are successfully performed and confirmed using a computation engine.Based on the symbolic solutions，actuator control and trajectory tracking are designed so that the kinematic response is proved to be correct and effective.

Fig.8 Adept motion for kinematic response.(a)Motion trajectory;(b)Driving angles of inverse kinematic

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