Wei－Guo Wu，Wen－Qian Du

(Humanoid and Gorilla Robot＆Its Intelligent Motion Control Lab，Harbin Institute of Technology，Harbin 150001，China)

1 Introduction

Legged-walking robot has more and more application in fields such as industry and daily life.Because of its convenience in work，especially in environment in which human are dramatically hard to be engaged，it will play a big role.And the walking stability is one of the most important characters which must meet specifications before robot is applied［1］.Therefore，legged-walking robot should be trained in lab，in which the training environment can change freely with 6-DOF.In this paper，the idea of applying specific platform in robot stability training is proposed for the first time.Therefore，such a new kind of mechanical platform should be designed and analyzed.When legged-walking robot is trained on the moving platform，the height and slope of every step foothold are different.So the robot should adjust its position and orientation to maintain balance.After training，the robot should achieve the ability of making adjustments on itself to maintain balance on uneven ground or on moving floor outside the lab.

Most of the existing training platforms have pure parallel or serial mechanism such as the flight simulator or vibration table.Pure serial mechanism have the advantages of big work space，flexible operation，easy forward kinematics and dynamics［2］，but also have the weaknesses of difficulty in solving inverse kinematics with analytical solution，low stiffness and low load capacity［3］.By contrast，pure parallel mechanism like Stewart［4－5］has better stiffness，higher load capacity，smaller inertia，improved weight/load ratio，averaged position-stance error［6－7］，higher positioning accuracy and easier inverse kinematics［8－9］，meanwhile with the weaknesses of smaller work space，more complex structure， hard forward kinematics［10－11］and dynamics［12－13］

The mixture style，serial-parallel mechanism，can combine both strengths consisting of high carrying capacity， big work space and other characteristics［14－16］.However，most of the existing serial-parallel platforms havesmall platform，high structure and other characters［17－19］.Therefore，for the function of stability training of robot，a new suitable kind of platform needs to be proposed.

2 Problem Proposed

For the purpose of stability training of leggedwalking robot，the robot should be able to walk up onto the platform from the floor and moves with the moving platform.Therefore，the platform is supposed to be lower than 0.5 m in height，and should be convenient for observation and operation.One step of robots withthe equivalent size of human，such as P3 or ASIMO，is assumed to be 0.6 m long.In order to enable robots to walk several steps and turn around on the platform，training platform should have enough space such as 3 m×2.5 m and have no ceiling.What’s more，the platform must have rational structure，good stiffness，high load capacity and big work space.

Fig.1 shows how to get legged-walking robot trained on the platform.When legged-walking robot is trained on the platform，the sensor of the platform brings the signal of the position and orientation of the platform back to the robot control system.And there are sensors at every robot joint，whose function is to transfer the signal of rotational displacement，force and moment to the balance correction generator of the robot，so that robot can adjust its body to maintain balance.

Fig.1 Concept schematic diagram of stability training system for legged-walking robot

3 Design of 6-DOF Platform

3.1 Mechanism

Selected in serial，parallel and serial-parallel mechanism，the third program is chosen and designed to be a two-tier mechanism as shown in Fig.2.In this design，there are n=13 rigid bodies and g=16 joints，consisting of 7 prismatic pairs and 9 spherical joints.The DOF of designed platform can be calculated by the following formula:

For all prismatic pairs where i=1－4，14－16，fi=1 and for all spherical joints where i=5－13，fi= 3.Because rods between two spherical joints can rotate around their own axis，there are 4 local DOF，which means flocal=4.According to the above formula，up board 18 has 6 degrees of freedom.

Mechanism principle:four parallel prismatic pairs 1－4 are driven by servo motors separately and connected with four rods by spherical joints 5－8.The rods are connected with the down board 17 by spherical joints 9－12.In the initial state，rectangle formed by center points of spherical joints 5－8 is bigger than the one formed by center points of spherical joints 9－12，so that the structure is like an inverted trapezoid body rather than a parallel rectangular body.And the lengthwidth ratios of the two rectangles are different.Down board 17 connects with the spherical joint 13 at the central point and spherical joint 13 can move up and down along the vertical rail 14.

The rail of prismatic pair 15 connects solidly with down board 17 and its slider is connected with the rail of prismatic pair 16 solidly.The slider of prismatic pair 16 is connected to the serial board 18 solidly.Prismatic pairs 15，16 are driven by servo motors.

Fig.2 Schematic diagram of the 6-DOF serial-parallel mechanism

3.2 Mechanical Structure

As shown in Fig.3，down tier applies parallel mechanism composed by four liner units，whose sliders connect with down board by rods whose two ends are spherical joints.There are X and Y liner units between up and down tiers，and thus the two-tier structure forms the serial-parallel mechanism.As shown in Fig.4，there is a hole with bigger size than the Y liner unit in the center of down board 17，so that X and Y liner units can be installed inside the hole and the height of up board 18 is lower.X liner unit is connected to the down board 17 solidly through support plate and support flange.There are vertical raceways within the corresponding surfaces of up and down boards in which there are balls to form the rolling support which can improve the support rigidity.

Fig.3 3-dimensional assembly structure

As shown in Fig.4(a)，4 auxiliary supports are installed in corners of down boards，to support weight of up and down boards and their components.The four auxiliary supports can change their length and angle as down board 17 moves and are designed as universal multi-section extendable rods，in which there are compression springs.

Fig.4 Rolling support within boards and universal multisection extendable rod

X liner unit connects with the support flange，which connects with center rail by central spherical joint.Center rail plays the role which makes down board 17 not to be able to move along axis X and Y. Therefore，down board 17 has 4 DOF，including moving along axis Z and rotating around axis X，Y and Z.Up board 18 can move relatively to down board 17 in X and Y direction，so that finally it achieves the function of movement with 6 degrees of freedom.

4 Kinematics，Dynamics and Error Analysis

4.1 Inverse Kinematics

In this part，the formula of 6 motors’rotational displacement is derived by using the position and orientation of up board.

As shown in Fig.3，several coordinate systems are established，including base coordinate system O－XYZ，down board moving coordinate system Op－XpYpZpand up board moving coordinate system Os－XsYsZs.

Transformation matrix of Op－XpYpZprelative to O－XYZ is as follows:

Assuming that the length of rods with spherical joints is hi=h and down spherical joint center point is note as a'iand its coordinate value is(a'ipx，a'ipy，0)in Op－XpYpZpand(a'ix，a'iy，a'iz)in O－XYZ.Similarly，assuming that up spherical joint center point is note as Ai'and its coordinate value is(A'ix，A'iy，A'iz)in O－XYZ and A'iz=li，where liis the height of up spherical joint centers in O－XYZ.The distance between a'iand A'iis as follows:

Transformation matrix of Os－XsYsZsrelative to Op－XpYpZpis written as Eq.(3).

So transformation matrix of Os－XsYsZsrelative to O－XYZ has the form of Eq.(4).

Coordinate value of Osrelative to O－XYZ becomes

Combine Eqs.(2)and(5)，we can get l1－l6.

The relationship between motors’rotational displacement qiand sliders’displacement can be written as Eq.(9)，

where l is initial position of sliders and P is screw pitch.

4.2 Differential Inverse Kinematics

In this part，the formula of 6 motors’rotational velocity will be derived by using the velocity and rotational velocity of up board.The velocity and rotational velocity of Oprelative to O－XYZ has the form of Eqs.(10)and(11).

The velocity of a'iin O－XYZ can be written as

The velocity of parallel slider equals to the projection of vaion the axis direction of parallel liner unit lni，so we have

The velocity of Osin Op－XpYpZpis as Eq.(16).

The velocity of Osin O－XYZ can be written as

Therefore the sliders’velocity of X and Y liner units are obtained as follows:

where the parameters can be written as

Combining Eqs.(14)，(15)，(18)with Eq.(19)，we can get Eq.(20).

The relationship between motors’ rotational velocity˙qiand sliders’velocity˙liis shown in Eq.(21).

Combining Eq.(20)with Eq.(21)，we can get

4.3 Forward Kinematics

Asisknown，forward kinematicsofparallel mechanism is relatively harder to be derived because of the necessity to solve nonlinear equations.Substitution method is applied in this paper.Transformation matrix of down board can be written as Eq.(23)，and Eq.(24) shows the relationship between parameters in Eq.(23).

Substituting Eq.(23)into Eq.(2)，we can get Eq.(25).

We can get the position and orientation of down board shown in Eqs.(26)－(29)by combining Eq.(24)with Eq.(25)and keep doing substitution and arrangement.

In Eqs.(26)－(29)，the parameters are shown as follows:

According to Eqs.(26)－(29)，Tpcan be obtained，and then combined with Eq.(5)，position and orientation of the up board in coordinate system O－XYZ can be obtained.

4.4 Differential Forward Kinematics

Combining Eqs.(11)，(12)with(22)，the velocity and rotational velocity of up board can be represented as Eq.(30).

In Eq.(30)，J is the Jacobian matrix.

4.5 Dynamics Analysis

In this part，Lagrange method is applied to analyze dynamics of the platform.

where τ represents general forces relative to q.

The platform is divided into five parts.In each part，velocity and rotational velocity are equal and have same orientation.

1)Up board and components fixed to it;

2)Down board and components fixed to it;

3)Y liner unit and its follow-up system;

4)Four rods with spherical joints at both ends;

5)All spherical joint units.

The translational and rotational energy of each part are not hard to be derived，so we do not list them here.

Then dynamics formula can be derived as Eq.(32)，

where M(q)s inertia matrix;G(q)s gravity matrix; and V(q，˙q)scoriolisforce and centrifugalforce matrix.

Then the relationship between τiand qi，˙qi，¨qican be derived by combining Eqs.(31)and(32)and forward kinematics with differential forward kinematics.

4.6 Error Analysis

In this section，the relationship between position and orientation errors of up board and all structures as well as movement errors will be established.

Vector relationship of the parallel part is as follows:

where Biis the coordinate value of the projection point of A'ion the plane XOY in coordinate O－XYZ.

By doing differentiate on both sides of Eq.(33) and arrangement，the error formula of parallel part can be written as Eq.(34)，

where δrp= [δCxδCyδCz+δPz]Tis down board position error;δθ= [δθxδθyδθz]Tis orientation error of both up and down board;δC is error of sphericaljoint13;ui=lni;and δmi= [δ a'ipδBi]Tis error of spherical joints 5－12.

According to Eq.(34)，[δPδθ δθ δθ]Tzxyzbecomes

Then δrpand δθ are obtained.

Similarly vector relationship of the serial parts can be written as

where uzis unit normal vector of up board in O－XYZ.

Similar to parallel part，serial part error formula is as follows:

where δr= [δx δy δz]Tis position error of up board.

The error formula of up board is obtained by combining Eqs.(33)－(36).For the future use，the error model is benefit for structure and control system design.

5 Simulation

3D model of the platform is built with ADAMS software，and the amplitude limiting random motions are applied to the up board.In the simulation，the parallel and serial sliders move passively，F(xiàn)ig.5 shows one moment of the simulation.A 100 kg robot is fixed with the upper board during the simulation to test load capacity of the platform.When it is used to train the stability of robot，the robot just stand on the up board，rather than fixed with the up board.The angular displacement curves and angular velocity curves of parallel unit motors are shown in Fig.6.

Fig.5 Simulation in ADAMS

Fig.6 Angular movement of 4 parallel unit motors

The angular displacement curves and angular velocity curves of serial unit motors are as shown in Fig.7，and the movement curves of up board are shown in Fig.8.

Fig.7 Angular movement of 2 serial unit motors

Fig.8 Movement curves of up board center

The curves of Fig.6 are saved and are imported back as drive curves of parallel unit motors and the drive of Opare removed at the same time.The simulation movement curves of up board achieved are the same as those in Fig.8.

To ensure the correctness of kinematics derivation，for example，taking the time point t=30，up board position，orientation，velocity and angular velocity are taken from Fig.8 and are shown in Tables.1 and 2.

Table 1 Up board position and orientation

Table 2 Up board velocity and angular velocity

The parameters’data of Tables 1 and 2 are substituted into the models of inverse kinematics and differential inverse kinematics.And 6 motors’angular displacement and angular velocity can be achieved as shown in Tables 3 and 4.

Table 3 6 motors’angular displacement (°)

Table 4 6 motors’angular velocity((°)/s)

The calculated results are the same as those in Fig.6.Then the data of Tables 3 and 4 are taken as inputs，through the model of forward kinematics and differential forward kinematics，the outputs are the same as those in Tables 1 and 2.

Therefore，the inverse and forward kinematics can be derived reversibly and they match the simulation result，which certifies the correctness of kinematics analysis.

Drive torque of 6 motors are shown in Fig.9.It can be found that the parallel drive torque on the diagonal of the virtual prototype are almost the same，that is τ1≈τ3，τ2≈τ4.

According to the simulation results，the 4 parallel drive torque needed is about 140 N·m and the 2 serial drive torque needed is about 0.8 N·m.Taking auxiliary support between down board and floor and resistance like friction into consideration，parallel and serial torque needed should be 160 N·m and 1 N·m.

Apply SGMGH SGMCS 2ZN A as parallel drive unit motors and SGMGH SGMAV 06A as serial drive unit motors.Rexroth TKK 30－325 AI is taken as liner units，whose diameter×pitch=32 mm×32 mm.Therefore，parallel sliders can move at 80 mm/s and serial sliders can move at 1600 mm/s.

Fig.9 Drvie torque of 6 liner unit motors

6 Conclusions

A new kind of 6-DOF serial-parallel platform is proposed which is suitable for stability training of legged-walking robot.It integrates both advantages of serialplatform and parallelplatform in support capacity，work space and kinematics.It also has low and big platform，no ceiling，and is convenient for observation and operation.This platform can move along axis X，Y，Z and rotate around axis X，Y，Z. Rolling support is designed between up board and down board to improve the support strength.And four universal multi-section extendable rods are designed between the floor and down board as auxiliary support to reduce the drive moment of parallel unit motors.

The formula is derived between up board position and orientation and 6 motors’rotational movement; especially the analytical solutions of forward kinematics model are obtained，which makes solving process fast.Error analysis model of up board is established，which makes it easily find main influenced position-stance errorsand main system errorsources.Dynamics simulation is done，which ensures the platform has a load capacity of 100 kg，and kinematics simulation is done to ensure the correctness of the kinematics derivation and finally ensure the feasibility of this design.

The designed platform has already applied for patent.The future research is about the legged-walking robot stability training.Through establishing control algorithm，after several times’training on the platform with free movement，the robot can achieve the ability of adjusting on its own to adapt to other unstable environment outside the lab.

［1］Wei Hui，Shuai Mei，Wang Zhongyu.Dynamically adapt to uneven terrain walking control for humanoid robot.Chinese Journal of Mechanical Engineering，2012，25(2):214－222.

［2］ Arakelian V，Sargsyan S.On the design of serial manipulatorswith decoupled dynamics.Mechatronics，2012，22(6):904－909.

［3］ZHAO Yongjie.Inverse displacement analysis of the general six degree-of-freedom serial robot based on optimization method.Journal of Chongqing University(English Edition)，2011，10(2):60－67.

［4］Dasgupta B，Mruthyunjaya T S.The Stewart platform manipulator:a review.Mechanism and Machine Theory，2000，35(1):15－40.

［5］Yao Rui，Zhu Wenbai，Huang Peng.Accuracy analysis of Stewart platform based on interval analysis method.Chinese Journal of Mechanical Engineering，2013，26(1):29－34.

［6］Cheng Gang，Ge Shirong，Wang Yong.Error analysis of three degree-of-freedom changeable parallelmeasuring mechanism.Journal of China University of Mining and Technology，2007，17(1):101－104.

［7］Kim H S，Choi Y J.Kinematic error bound analysis of the Stewart platform.Journal of Robotic Systems，2000，17 (1):63－73.

［8］Hu Ming，Shi Jiashun.The kinematic analyses of the 3-DOF parallelmachine tools.InternationalJournalof Automation and computing，2011，8(1):107－111.

［9］Lu Yi，Zhang Xuili，Sui Chunping，et al.Kinematics/ statics and workspace analysis of a 3-leg 5-DoF parallel manipulator with a UPU-type composite active constrained leg.Robotica，2013，31(2):183－191.

［10］Qi Ming，Qie Yanhui.Forward kinematics analysis for a Novel5-DOF papallelmechanism using tetraheadron configuration.Chinese Journal of Mechanical Engineering，2007，20(6):1－4.

［11］Choi H B，Konno A，Uchiyama M.Closed-form forward kinematics solutions of a 4-DOF parallel robot.International Journal of Control，Automation and Systems，2009，7(5): 858－864.

［12］Gao Feng，Wang Guofu，Cui Ying，et al.Dynamic modeling and analysis of 9-DOF omnidirectional legged vehicle.Chinese Journal of Mechanical Engineering，2011(4): 515－521.

［13］Hao Qi，Guan Liwen，Wang Jinsong，et al.Dynamic feedforward control of a novel 3-PSP 3-DOF parallel manipulator.Chinese Journal of Mechanical Engineering，2011(4):676－684.

［14］Hu Bo，Lu Yi，Tan Qing，et al.Analysis of stiffness and elastic deformation of a 2(SP+SPR+SPU)serial—parallel manipulator. Robotics and Computer-Integrated Manufacturing，2011，27(2):418－425.

［15］Garcia-Murillo M，Gallardo-Alvarado J，Castillo-Castaneda E.Finding the generalized forces of a series-parallel manipulator.International Journal of Advanced Robotic Systems，2013，10:1－10.

［16］Tanev T K.Kinematics of a hybrid(parallel—serial)robot manipulator.Mechanism and Machine Theory，2000，35 (9):1183－1196.

［17］Wang X，Baron L，Cloutier G.Topology of serial and parallel manipulators and topological diagrams.Mechanism and Machine Theory，2008，43(6):754－770.

［18］Gallardo-Alvarado J，Rodríguez-Castro R，Aguilar-Nájera C R，et al.A novel six-degrees-of-freedom series-parallel manipulator. Journal of Mechanical Science and Technology，2012，26(6):1901－1909.

［19］Xie F，Liu X，Zhang H，et al.Design and experimental study of the SPKM165，a five-axis serial-parallel kinematic milling machine.Science China Technological Sciences，2011，54(5):1193－1205.