Research ArticleAnalysis and Optimization of Interpolation Points forQuadruped Robots Joint Trajectory

Mingfang Chen , Kaixiang Zhang , Sen Wang , Fei Liu, Jinxin Liu,and Yongxia Zhang

Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming, Yunnan, China

Correspondence should be addressed to Sen Wang; wangsen0401@126.com

Received 27 April 2020; Accepted 25 May 2020; Published 1 July 2020

Guest Editor: Qiang Chen

Copyright © 2020 Mingfang Chen et al. +is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Trajectory planning is the foundation of locomotion control for quadruped robots.+is paper proposes a bionic foot-end trajectorywhichcan adapt to many kinds of terrains and gaits based on the idea of trajectory planning combining Cartesian space with joint space.Trajectory points are picked for inverse kinematics solution, and then quintic polynomials are used to plan joint space trajectories. In orderto ensure that the foot-end trajectory generated by the joint trajectory planning is closer to the original Cartesian trajectory, the dis-tributions of the interpolation point are analyzed from the spatial domain to temporal domain. An evaluation function was established toassess the closeness degree between the actual trajectory and the original curve. Subsequently, the particle swarm optimization (PSO)algorithm and genetic algorithm (GA) for the points selection are used to obtain a more precise trajectory. Simulation and physicalprototype experiments were included to support the correctness and effectiveness of the algorithms and the conclusions.

1. Introduction

Trajectory planning is the first step of quadruped robotstowards complex environments [1, 2]. At present, the tra-jectory planning is mainly divided into two categories:Cartesian space trajectory planning and joint space trajec-tory planning [3, 4]. +e former focuses on the macro area,while the latter pays attention to the micro area. +e Car-tesian space trajectory planning involves the actual motiontrajectory of actuator in three-dimensional space, while thejoint space trajectory planning studies the connectionproblem of joint angles in the process of motion [5]. Hence,for foot-end trajectory planning of quadruped robots,combining the above two methods can lay a superiorfoundation for its flexible movement in real time.

For the purpose of making the foot-end perform moreflexible in Cartesian space, Sakakibara et al. proposed a com-pound cycloidmethod and planned a smooth foot trajectory fortheir walking robot [6]. Based on their method, Wang et al.derived a zero-impact trajectory and the cycle control strategyfor one-leg gait [7]. On the other hand, Chevallereau et al.proposed a ballistic trajectory [8], which followed the principleof alternation of muscle movements. Vundavilli et al.

constructed a foot-end trajectory based on a cubic polynomialand planned an optimal gait for dynamic balance of bipedrobots [9]. In order to make the foot-end trajectory more bi-ological, Kim et al. proposed an elliptical trajectory to simulatethe backswing and retraction characteristics of animals whenthey departed or touched the ground [10]. Semini et al. addedthe foot speed in the flattened part of the semi-ellipse to co-ordinate the forward speed of the trunk during the trot motion[11]. Rong et al. used a compound trajectory consisting of cubiccurves with straight lines on their Scalf robot [12] and dem-onstrated a foot-end trajectory consisting of five-degree poly-nomials with four-degree polynomials in Scalf afterward [13].For more flexibility, the researchers of MIT used the Béziercurve in its MIT Cheetah [14], which utilized twelve positioncontrol points for the foot-end trajectory and connected themwith eleven-degree Bernstein polynomial curve. +e works ofthe these scholars have provided rich theoretical foundationsfor the research of the foot-end trajectory of the legged robotsand have developed the trajectory in the direction of highsmoothness, high bionics, and strong adaptability. However,the trajectory planning in Cartesian space requires repeatedinverse kinematics calculations during movement. As thecomplexity of the foot-end trajectory increases, it will cost a

HindawiComplexityVolume 2020, Article ID 3507679, 17 pageshttps://doi.org/10.1155/2020/3507679

mailto:wangsen0401@126.comhttps://orcid.org/0000-0002-3323-8168https://orcid.org/0000-0003-4922-899Xhttps://orcid.org/0000-0002-7392-578Xhttps://orcid.org/0000-0003-3676-8938https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/3507679

large computational load on the controller. Consequently, inthe process of movement, if the accuracy of the trajectory isguaranteed, the numbers of trajectory transition points willincrease subsequently, but the real-time performance of thelocomotion will be reduced. +erefore, joint space trajectoryplanning can alleviate the contradiction between motion ac-curacy and real-time in the Cartesian space trajectory planningto some extent.

Joint space trajectory planning is a way for locomotion viasampling the designed trajectory and extracting few trajectoryintermediate points to execute inverse kinematics solution.And then some smooth curves are used to connect the in-termediate points of the joint angles inversely solved by ki-nematics [15]. For getting the smooth connections betweenthe intermediate points, Liu et al. proposed a method forgenerating joint trajectories based on X-splines and quarticpolynomials [16]. In the Robotics Toolbox, Corke utilized afifth-degree polynomial to perform point-to-point joint spacetrajectory planning and obtained a smooth curve between twopoints [17]. Dehghani et al. also used this method to controlthe action of the five-link biped, ensuring continuous angularvelocity and angular acceleration during locomotion [18]. Inorder to make the robot’s locomotion process more stable,Gasparetto et al. endeavored to optimize the joint motiontime and jerkiness to ensure that the trajectory has sufficientsmoothness [19]. Liu et al. used a combination of Cartesianspace multi-order spline function and joint space multi-orderB-spline function to ensure continuous jerk while improvingthe smoothness of trajectory tracking control [20]. Zhonget al. performed 8-degree polynomial trajectory planning injoint space on the basis of 8-degree polynomial trajectoryplanning for the center of mass to form a correspondingrelationship [21]. In order to avoid the dynamic singularityduring the movement, Wang et al. parameterized the jointtrajectory into a fifth-order Bézier curve and combined theparticle swarm optimization (PSO) to optimize the curvecontrol points [22]. +e researches of the above scholars laidthe foundation for joint space trajectory planning, whichmade it perform better in practical applications. Accordingly,combining the trajectory planning methods of Cartesianspace with joint space may lead to better trajectory planningactual effects. However, regarding how to coordinate theCartesian trajectory and the joint trajectory, especially withregard to the choice of middle points, there is less researchwork done. Secondly, the errors between the actual foot-endcurve generated by the joint trajectory motion and thedesigned Cartesian trajectory need to be further reduced.

In this paper, a bionic foot-end trajectory was designedfor the physical quadruped robot platform namedMQ robotas shown in Figure 1. +e trajectory imitates the actualanimal movement laws and divides the foot-end trajectoryinto three segments: backswing, advance, and retraction. Inorder to meet the real-time requirements in actual use, jointspace trajectory planning was further performed. Eleven

points were picked up from the Cartesian trajectory forinverse kinematics solution, and fifth-degree polynomialswere used for piecewise interpolation. In order to explore theeffect of different points on the generated actual foot-endtrajectory, the intermediate points were taken at equal in-tervals distribution and equal arc length distributions forcomparing and analyzing. Furthermore, according to thecharacteristics of parameter function, the isochronouspoints and Chebyshev nodes were taken into the modes ofdistributions in the temporal domain [23]. Based on theanalysis results of the above modes, the PSO algorithm andgenetic algorithm (GA) are adopted to generate a betterdistribution of trajectory points, which use the errors be-tween the actual trajectory and the design trajectory as thefitness values by establishing the models of both trajectories.Further, simulation and prototype experiments were carriedon to verify the validity of the analysis and optimizationresults. +e innovations of this paper are as follows: (i) thefoot trajectory of the quadruped robots was divided intothree segments based on biological observation. (ii) Carte-sian space trajectory planning and joint space trajectoryplanning were combined in order to obtain a better practicalapplication effect. (iii) +e distribution of the interpolationpoints was analyzed and optimized to make the actualtrajectory closer to the original design trajectory.

2. Bionic Trajectory Planning ofQuadruped Robots

2.1. Cartesian Space Trajectory Planning. In the process ofanimal movements, the backswing and retraction of the feetare universal, which could ensure the continuity of itsmovement process [24]. As shown in Figure 2, according tothe characteristics during the actual movement of animals,this paper utilizes a bionic foot-end trajectory for thequadruped robots. +rough the trajectory decomposition,the foot-end trajectory of Cartesian space is decomposedinto parameter functions of the forward direction X and thelifting direction Y with the parameter t (time), and thenpiecewise quintic polynomial curves are used for interpo-lation. Among them, the piecewise curves in the X directionhave correspondingly added backswing and retractionsegments based on the forward segments proposed in [6, 7],and the piecewise curve in the Y direction maintains thecharacteristics of symmetry. As deduced from quinticpolynomials, the trajectory can be gotten by equations (1)and (2). Taking advantage of the quintic polynomial curvewhich needs six parameters to complete itself, the trajectoryshould give six initial elements to deduce its form, so that theendpoint position, speed, and acceleration of each curve areindependently adjustable to meet the requirements of dif-ferent terrains and ensure the locomotion processsmoothness without impact theoretically:

2 Complexity

12SS1 − 6vs0 · TS1 − as0 − as1( · TS12 2TS15

· t − t0( 5

+−30SS1 + 16vs0 · TS1 + 3as0 − 2as1( · TS12

2TS14· t − t0(

4

+20SS1 − 12vs0 · TS1 − 3as0 − as1( · TS12

2TS13· t − t0(

3+12as0 · t − t0(

2+ vs0 · t − t0( + s0, t ∈ t0, t1

12SS2 − as1 − as2( · TS22 2TS25

· t − t1( 5

+−30SS2 + 3as1 − 2as2( · TS22

2TS24· t − t1(

4

+20SS2 − 3as1 − as2( · TS22

2TS23· t − t1(

3+12as1 · t − t1(

2+ s1, t ∈ t1, t2

12SS3 − 6vs d · TS3 − as2 − as d( · TS32 2TS35

· t − t2( 5

+−30SS3 + 14vs d · TS3 + 3as2 − 2as d( · TS32

2TS34· t − t2(

4

+20SS3 − 8vs d · TS3 − 3as2 − as d( · TS32

2TS33· t − t2(

3+12as2 · t − t2(

2+ s2, t ∈ t2, td .

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

As shown in equation (1), the definitions of the indirectparameters SS1, TS1, SS2, TS2, SS3, TS3 (obtained by calculationof direct parameters) are as follows:

SS1 � s1 − s0,

TS1 � t1 − t0,

SS2 � s2 − s1,

TS2 � t2 − t0,

SS3 � sd − s2,

TS3 � td − t2,

y �

12HH1 − 6vh0 · TH1 − ah0 − ahw( · TH12 2TH15

· t − t0( 5

+−30HH1 + 16vh0 · TH1 + 3ah0 − 2ahw( · TH12

2TH14· t − t0(

4

+20HH1 − 12vh0 · TH1 − 3ah0 − ahw( · TH12

2TH13· t − t0(

3+12ah0 · t − t0(

2+ vh0 · t − t0( + h0, t ∈ t0, tw

12HH2 − 6vh d · TH2 − ahw − ah d( · TH22 2TH23

· t − tw( 5

+−30HH2 + 14vh d · TH2 + 3ahw − 2ah d( · TH22

2TH24· t − tw(

4

+20HH2 − 8vh d · TH2 − 3ahw − ah d( · TH22

2TH23· t − tw(

3+12ahw · t − tw(

2+ hw, t ∈ tw, td ,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where the indirect parameters HH1, TH1, HH2, TH2, aredefined as follows:

Complexity 3

HH1 � hw − h0,

TH1 � tw − t0,

HH2 � hd − hw,

TH2 � td − tw.

(3)

For the direct parameters (boundary conditions) inequations (1) and (2), the sizes of time ti and positions si, hiare adjustable according to the environmental terrains, andthe speeds vsi, vhi and accelerations asi, ahi are adjustedaccording to the robot’s forward speed. In this paper, for

studying the joint space trajectory planning of the foot-endtrajectory when crawling on the ground, the values of theabove direct parameters are set as shown in Table 1.

In Table 1, the positions can be adjusted according to theneeds of terrains and step length. s0 and h0 are the startingpoints of the motion, which coincide with the origin of thecoordinate system by default. In actual use, translation canbe performed according to the difference in the set coor-dinate system. +e foot-end trajectory after translation isshown in Figure 3. +e speed values at t1, t2, and tw are 0 bydefault, so they do not appear in equations (1) and (2). +espeed values at t0 and td refer to the speed in [7] for zero

Body

HFE joint

HAA joint

KFE joint HFE motor KFE motor

HAA motor

Figure 1: Experimental prototype of MQ quadruped robot.

X

Y

Trot

Canter

Foot-end retraction can reduce the landing impact

when touchdown

Foot-end backswing can reduce the commutation

impact of joint

BackswingRetraction

Figure 2: +e backswing and retraction of feet during animal movements [24–27].

4 Complexity

impact, and the velocity curves of the submotion in twodirections are shown in Figure 4. +e values of as1, as2, andasw are optimized to obtain smooth acceleration curves asshown in Figure 5.

In Figure 3, the starting point in the X direction is shiftedto the right by 15(mm) to facilitate the representation of thefoot-end trajectory. In the same way, different translationcan be performed on the starting points for different co-ordinate systems.

2.2. Kinematics Analysis. In order to carry on the joint spacetrajectory planning in the next discussion, the single-legstructure of the experimental prototype needs to be simplified,

and then the kinematics analysis can be performed. As shownin Figure 6, the single-leg structure of the prototype is extractedand compared with the general platform. It is worth men-tioning that many researchers had studied the kinematicssolution of the general platform [28–30]. On these bases, therelationship between the single-leg drive angle of the prototypeand the general platform is shown in the following:where θ1,θ2 , and θ3 are the driving joint angles of the actual prototype.Respectively, θ1, θ2, and θ3 are the joint angles of the generalplatform.+e other parameters involved are the dimensions ofthe legs of quadruped robot, as shown in Table 2.

θ1 � θ1, (4)

θ2 � θ2, (5)

θ3 � −180 + θ2

+ arccos2a24 − 2a4 · l2 · cos θ3 − 10(

2a4 ·�������������������������a24 + l

22 − 2a4 · l2 · cos θ3 − 10(

+ arccos2l22 − 2a4 · l2 · cos θ3 − 10(

2l2 ·�������������������������a24 + l

22 − 2a4 · l2 · cos θ3 − 10(

,

(6)

2.3. Joint Space Trajectory Planning. Due to the limitedcomputation of the controller [31], in order to alleviate thecontradiction between the accuracy of foot-end trajectory and

Table 1: Boundary conditions of the bionic trajectory.

Parameters Forward direction X Lifting direction YTime (s) t 0 � 0 t 1 � 0.25 t 2 � 0.75 t d � 1 t 0 � 0 tw � 0.5 t d � 1Position (mm) s 0 � 0 s 1 � −15 s 2 � 85 s d � 70 h 0 � 0 hw � 61.8 h d � 0Velocity (mm/s) vs0 � 0 0 0 vs d � 0 vh0 � 0 0 vh d � 0Acceleration (mm/s2) a s0 � 0 a s1 � 1760 a s2 � −1760 a sd � 0 a h0 � 0 a hw � −1648 a hd � 0

20 40 60 80 1000X (mm)

0

10

20

30

40

50

60

70

Y (m

m)

Figure 3: Foot-end bionic trajectory in Cartesian space.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time (s)

X directionY direction

–300

–200

–100

0

100

200

300

400

Vel

ociti

es (m

m/s

)

Figure 4: Velocity curve of submovement of trajectory.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time (s)

Acc

eler

atio

ns (m

m/s

2 )

X directionY direction

–2000

–1500

–1000

–500

0

500

1000

1500

2000

Figure 5: Acceleration curve of submovement of trajectory.

Complexity 5

the real-time of locomotion, the joint space trajectory planningis carried out for the foot-end trajectory. +e movementprocess of the entire trajectory is divided into a small number ofn segments which leads to n+1 points being selected as in-terpolation points. +erefore, it can reduce the computationalburden of the controller to a large extent.

+e selections of different interpolation function have anessential influence on the effect of joint space trajectoryplanning. A good interpolation function can reduce theamount of calculations and improve the accuracy of themovement. At present, there are two kinds of interpolationmethods commonly used: high-order interpolation andpiecewise low-order interpolation. +e high-order interpola-tion has a large amount of computation, and it is easy toproduce the Runge phenomenon which will lead to the lack ofaccuracy in the beginning and end stages of motion [32].Segmented low-order interpolation can avoid the occurrence ofRunge phenomenon, but due to the order limit, the motioncharacteristics of each interpolation point cannot be adjustedindependently. +erefore, counterpoising the advantages anddisadvantages of the two methods, the joint space trajectoryplanning is carried out by the method of piecewise quintic

polynomial interpolation. It can ensure that the angle, angularvelocity, and angular acceleration of each interpolation pointare independently controllable, and can correspond with thetrajectory planning of Cartesian space in Section 2.1. So it isbeneficial to further improving the trajectory accuracy.

Since the motion characteristics of quintic polynomials ateach interpolation point can be adjusted independently, theboundary conditions of function curve at both endpoints areset as follows:

θ t0( � θ0,_θ t0( � ω0,€θ t0( � α0,

θ tf � θf,

_θ tf � ωf,

€θ tf � αf.

(7)

Taking time t as the independent variable, the functionexpression of angle θ(t) is illustrated as

RFLF

LHRH

RF

LF

RH

LH

Single-legged model

General platformExperimental prototype

Single-legged model

BridgeOther innovative quadruped robot

platforms

Leg structure analysis

Overall structure analysis

θ1

θ2

θ3

l1

l2

l3

Gx

y

F

H

C

D

A B

a1

a6

a2

a4a3a5

l2

l3

2l1

θ2(θ2)

θ1(θ1)

θ3

θ3

EE′

10°

Figure 6: Simplification and transformation between the single-leg structure of the prototype and the general platform.

Table 2: Properties of the prototype leg.

Parameter Description Value (mm)a1 Length of a1 50a2 (2l1) Length of a2 or 2l1 84a3 Length of a3 50a4 (a6) Length of a4 or a6 34a5 (l2) Length of a5 or l2 170l3 Length of l3 245

6 Complexity

θ(t) �12 ·Δθ − ω0 +ωf · 6 ·Δt − α0 −αf ·Δt2

2 ·Δt5t − t0(

5

+−30 ·Δθ + 16ω0 + 14ωf ·Δt + 3α0 − 2αf ·Δt2

2 ·Δt4t − t0(

4

+20 ·Δθ − 12ω0 + 8ωf ·Δt − 3α0 −αf ·Δt2

2 ·Δt3t − t0(

3

+α02

t − t0( 2

+ ω0 t − t0( + θ0, t ∈ t0, tf ,

(8)

where the indirect parameters Δθ, Δt in (8) are defined asfollows:

Δθ � θf − θ0,

Δt � tf − t0.(9)

It can be seen from equation (8) that the interpolationfunction needs to put in the boundary conditions θ0, θf, ω0,ωf, α0, αf and then to solve the six unknown parameters in theθ(t) expression according to these six known parameters.Accordingly, the unique solution of the equation is obtained.+erefore, the reasonable setting of the boundary conditionsθ, ω, α is the primary to solve this case. In order to ensurethat the function curve is smooth without fluctuation, theboundary conditions are solved using the method of dif-ference quotient and then set as their average value. Aftercalculating the difference quotient of the known angularvalues θk−1, θk, and θk+1 in order, the average value of theadjacent is taken as the angular velocity ωk, which isexpressed as

ωk �θk − θk−1( /Δtk−1 + θk+1 − θk( /Δtk

2, (10)

where Δtk is the time interval between the time tk corre-sponding to the interpolation point k and the time tk+1corresponding to the next interpolation point k+1, and Δtk−1is the same definition as Δtk. Similarly, the solution processof the angular acceleration αk is shown in the followingequation:

αk �ωk − ωk−1( /Δtk−1 + ωk+1 − ωk( /Δtk

2. (11)

For the start and end points of the whole trajectory, ωand α, the boundary parameters, can be specified accordingto the motion state and forward speed. For the crawling gait,the start and end boundary conditions of ω and α are set asω0 � 0, ωn � 0, α0 � 0, αn � 0. According to the above deri-vation method, the solution process of boundary conditionsat each interpolation point is shown in Figure 7.

3. SelectionandAnalysisof InterpolationPoints

Different interpolation functions can result in differenttrajectory planning effects; accordingly, selecting differentinterpolation points can also lead to different trajectoryaccuracy [33]. However, there are few discussions with theeffects caused by different interpolation points distributions.

In order to reduce the number of intermediate points injoint space trajectory planning and have better trajectoryreducibility, the selection of interpolation points is par-ticularly important. Appropriate interpolation points canreduce the number of points and increase the proximity tothe original trajectory [34]. Based on the above bionictrajectory, the whole trajectory is divided into 10 segmentsfrom the spatial domain and the temporal domain, so thatthe trajectory is symmetrical and the principle of fewerinterpolation points is also guaranteed. Accordingly, 11corresponding trajectory points are obtained. +en, theinverse kinematics solution is used to obtain the inter-polation points of joint trajectory.

3.1. Trajectory Division in Spatial Domain. According to thedesigned Cartesian trajectory, the equidistant and equal arclength are as the method to partition trajectory, and then thecorresponding joint trajectory interpolation points aresolved by inverse kinematics for analysis. Firstly, the forwarddirection X of the trajectory is divided into 10 equal parts,and 11 corresponding trajectory points are obtained by turnsas shown in Figure 8(a), and then the corresponding jointtrajectory interpolation points are obtained by kinematicinverse solution. In the same way, the trajectory points areobtained after dividing the trajectory by equal arc length, asshown in Figure 8(b), and then the inverse kinematics so-lution is carried out for them to acquire the interpolationpoints.

In Figure 8, the coordinates of 11 trajectory pointsobtained by uniform X are shown in Table 3, and the co-ordinates of 11 trajectory points obtained by uniform arc areshown in Table 4. In Figure 8(a), between points 1, 2, and 3,since X decreases first and then increases, there is inter-section between two adjacent intervals. Similarly, there isalso intersection between points 9, 10, and 11. Hence, thebeginning and end of the trajectory do not appear to beequidistant. However, the change trend and length ofequidistant can be obtained from Table 3.

Put the trajectory points in Tables 3 and 4 into equations(4)–(6) for inverse kinematics solution, getting the corre-sponding joint trajectory interpolation points, and then putthe obtained interpolation points into equations (10) and(11), getting the boundary angular velocity and accelerationof each point. +en put them into equation (8) for the joint

Point

θ:

ω:

α:

0 1 2 3

θ0 θ1 θ2 θ3

ω0 ω1 ω2

α0 α1

n – 2 n – 1 n

θn–2 θn–1 θn

ωn–2 ωn–1 ωn

αn–1 αn

Figure 7: Solution process of interpolation point boundarycondition.

Complexity 7

space trajectory planning. Finally, the overall joint anglecurves are shown in Figure 9.

In Figure 9, the black solid line is the original jointtrajectory corresponding to the original design Cartesiantrajectory, HFE joint is the hip flexion/extension joint, andKFE joint is the knee flexion/extension joint [35]. Bycomparing the joint trajectories obtained from the twotrajectory points’ distribution in Figure 9, it can be seen that

the joint trajectories generated by the two methods arebasically close to the original joint trajectory, and the trend isthe same. However, there are some errors in details, espe-cially in the beginning and end areas of the trajectories. Itcan be seen from the local magnifying figure at the rightbottom that the joint trajectory obtained by uniform arcdivision is closer to the original joint trajectory than that byuniform X division. On the other hand, it can be found fromFigure 8 that the distribution of trajectory points by uniformarc is more dense in the starting and ending areas. So, thereis a guessing that enhancing the points density in the startingand ending areas may help to increase the accuracy. Ac-cordingly, distribution of the trajectory points needs to befurther adjusted.

3.2. Trajectory Division in Temporal Domain. Owing to thebionic trajectory is a set of parameter equations abouttime t, it can directly divide the time to get the coordinatevalues of X and Y at the same time values, and then thecorresponding trajectory points can be obtained and thetrajectory can be divided. Referring to the current com-mon methods of parametric function interpolation, thechoosing of interpolation points in segmented splineinterpolation is usually by the methods of uniform pa-rameterization, chord length, centripetal model, etc. [36].On the other hand, in the interpolation of higher-orderpolynomial functions, in order to avoid Runge phe-nomenon, Chebyshev points are often used as interpo-lation points [23]. +erefore, referring to the abovemethods, the time t is divided into two modes: uniformtime and Chebyshev nodes. Accordingly, the corre-sponding coordinate values are solved, and then the in-verse kinematics is used to obtain the interpolation pointsand the joint space trajectory planning is performed, sothat the coordinate values corresponding to the iso-chronous points are shown in Table 5, and its distributionin the trajectory is shown in Figure 10(a).

20 40 60 80 1000X (mm)

TrajectoryUniform X

0

10

20

30

40

50

60

70Y

(mm

)

(a)

20 40 60 80 1000X (mm)

TrajectoryUniform arc

0

10

20

30

40

50

60

70

Y (m

m)

(b)

Figure 8: Spatial domain trajectory divisions. (a) Uniform X. (b) Uniform arc.

Table 3: Corresponding trajectory points of uniform X.

Point number Time (s) X position (mm) Y position (mm)1 0 15 02 0.1999 2 14.85423 0.3596 11 47.19064 0.4144 24 56.00355 0.4589 37 60.42166 0.5000 50 61.80007 0.5411 63 60.42168 0.5856 76 56.00359 0.6404 89 47.190610 0.8001 98 14.854211 1 85 0

Table 4: Corresponding trajectory points of uniform arc.

Point number Time (s) X position (mm) Y position (mm)1 0 15 02 0.1980 2.1459 14.52523 0.2940 1.7623 33.77964 0.3710 13.3457 49.26855 0.4380 30.6892 58.70086 0.5000 50 61.80007 0.5620 69.3108 58.70088 0.6290 86.6543 49.26859 0.7060 98.2377 33.779610 0.8020 97.8541 14.525211 1 85 0

8 Complexity

Chebyshev points are generated by Chebyshev polyno-mials, which are divided into Chebyshev polynomial of thefirst class and Chebyshev polynomials of the second class.Chebyshev points can be obtained by finding the zero pointor the extreme point of Chebyshev polynomials. Amongthem, Chebyshev polynomial of the first class can be ob-tained by [37]

T0(x) � 1,

T1(x) � x,

Tn+1(x) � 2xTn(x) − Tn−1(x),

(12)

where n� 0, 1, 2, . . ., N. Equation (12) has n+ 1 extremepoints (including endpoints) in [−1, 1], which can be ob-tained by

xk � coskπn

, (13)

where k� 0, 1, 2, . . ., n. Taking n� 10, eleven Chebyshevpoints can be obtained to divide time, thereby generating 11

corresponding trajectory points as shown in Table 6, and theschematic diagram of trajectory points distributions is asshown in Figure 10(b).

In Figure 10, it can be found that the distribution of thetrajectory points generated by the two modes is more densein the beginning and end segments, while the distribution ofthe trajectory points in the middle segment is sparse. Amongthem, the distribution based on Chebyshev nodes is the mostobvious. +e joint trajectory effects of two modes are shownin Figure 11.

By comparing Figure 9 with Figure 11, it can be seen thatthe effect of joint trajectory generated based on temporaldomain division is better than the joint trajectory divided byspatial domain on the whole and is closer to the original jointtrajectory. Compared with the two partition methods inFigure 11, it can be found that the trajectory generated by theuniform time distribution is closer to the original jointtrajectory.

4. Interpolation Points Optimization

4.1.OptimizationProcessAnalysis. On the basis of the aboveanalysis, in order to explore the optimal distribution oftrajectory points, the intelligent optimization algorithmsare used to optimize the positions to obtain the best tra-jectory reducibility. Among many intelligent optimizationalgorithms [38–40], the PSO [41] and GA [42] are chosendue to their fast convergence and robustness. +e primarypurpose of optimization is to make the generated trajectorycloser to the original design trajectory. +erefore, takingthe errors between the original Cartesian trajectory and theactual trajectory as the optimization goal, the fitnessfunction of the PSO algorithm is constructed as shown inequation (14), and the fitness function of GA as shown inequation (15):

0.2 0.25 0.3 0.35

–0.92–0.9

–0.88–0.86–0.84–0.82

–1

–0.9

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3A

ngle

s (ra

d)

0.2 0.4 0.6 0.8 10Time (s)

Ideal curveCurve by uniform XCurve by uniform arc

(a)

0.25 0.3 0.35 0.4

–2.85–2.8

–2.75–2.7

0.2 0.4 0.6 0.8 10Time (s)

–3

–2.8

–2.6

–2.4

–2.2

–2

–1.8

Ang

les (

rad)

Ideal curveCurve by uniform XCurve by uniform arc

(b)

Figure 9: Joint trajectory effect based on spatial divisions. (a) HFE joint trajectory. (b) KFE joint trajectory.

Table 5: Corresponding trajectory points of uniform time.

Point number Time (s) X position (mm) Y position (mm)1 0 15 02 0.1 11.5056 2.52473 0.2 1.9952 14.87164 0.3 2.2816 35.06285 0.4 20.1888 54.00176 0.5 50 61.80007 0.6 79.8112 54.00178 0.7 97.7184 35.06289 0.8 98.0048 14.871610 0.9 88.4944 2.524711 1 85 0

Complexity 9

fp �T

n

n

i�1

������������������

xi − xi( 2

+ yi − yi( 2

, (14)

fg �1

(T/n) ni�1������������������

xi − xi( 2

+ yi − yi( 2

, (15)

where xi and yi are the Cartesian space coordinate valuesobtained from the positive kinematics solution of the jointtrajectory, xi and yi are the original Cartesian trajectorycoordinate values corresponding to xi and yi, T is the totalmotion time, and n is number of sampling points.

Let time t be the independent variable, the randompoints’ distributions as the initial positions. By continuouslyupdating the trajectory points’ positions and assessing thefitness, the distributions of trajectory points are graduallyoptimized until the convergence conditions are met. +isprocess can be represented by the flowchart as shown inFigure 12 below. Among them, (a) is the PSO flowchart and(b) is the GA flowchart.

In the process as shown in Figure 12, in order to obtainthe current individual’s fitness, the positive kinematicscalculation needs to be performed after the joint spacetrajectory planning to obtain the actual foot-end motion

trajectory. And then the error analysis based on equations(14) and (15) is executed to return the fitness value bycomparing with the original Cartesian trajectory. In thisprocess, there are multiple kinematic solving steps, so thecombination of PSO or GA function with Simulink is usedfor optimization. As shown in Figure 13, the fitness cal-culation system is established through Simulink, and thePSO or GA function is responsible for calling the Simulinkmodel to calculate the fitness, and then the current indi-vidual positions are adjusted to further optimize based onthe returned fitness value obtained by the Simulink model.

In Figure 13, “Points_in” is used to call the multidi-mensional individual positions which are the global vari-ables in the workspace generated by the PSO function, andthen they are put into “Interpolation_model” to calculate theboundary conditions of each interpolation point. Further-more, the corresponding piecewise quintic polynomial in-terpolation function is generated to perform joint spacetrajectory planning. +e “Forward-kinematics” are used toobtain the actual foot-end trajectory by forward kinematicssolving of the generated joint trajectory. +e actual foot-endtrajectory and the original trajectory generated in “Tra-jectory_function” are put into “Error_analysis” to analyze theerror between them by equations (14) and (15) and obtainthe fitness. Finally, the fitness is returned to the PSO or GAfunction through “Out_fitness” for next iteration.

4.2. Optimization Results Analysis. +rough the above PSOand GA method, the final obtained distributions of thetrajectory points are shown in Tables 7 and 8. From these twotables, we can find that the distributions obtained by the twomethods are relatively close, and both are further optimizedon the basis of the isochronous distributions.

In order to more comprehensively evaluate the closenessbetween the actual trajectory and the design trajectory, theintegrated absolute error (IAE) and integrated square error(ISDE) [43] are used as the performance indexes. +ey areshown as follows:

20 40 60 80 1000X (mm)

TrajectoryUniform time

0

10

20

30

40

50

60

70Y

(mm

)

(a)

14.8 150

0.10.2

20 40 60 80 1000X (mm)

TrajectoryChebyshev node

0

10

20

30

40

50

60

70

Y (m

m)

(b)

Figure 10: Temporal domain trajectory divisions. (a) Uniform time. (b) Chebyshev nodes.

Table 6: Corresponding trajectory points of Chebyshev nodes.

Point number Time (s) X position (mm) Y position (mm)1 0 15 02 0.0245 14.9209 0.04553 0.0955 11.8685 2.22734 0.2061 1.5635 15.94865 0.3455 8.3703 44.48376 0.5 50 61.80007 0.6545 91.6297 44.48378 0.7939 98.4365 15.94869 0.9045 88.1315 2.227310 0.9755 85.0791 0.045511 1 85 0

10 Complexity

IAE � |e(t)|dt, (16)

ISDE � (e(t) − e(0))2dt, (17)

where e(t)�������������������

(xi − xi)2 + (yi − yi)

22

and e(0) represents theaverage error. +e distributions of relevant trajectorypoints in Section 2 are compared, and the different per-formance index values of these distribution modes areshown in Table 9.

According to the analysis of the data in Table 9, the errorsof the trajectory generated by dividing time are overall fewerthan those generated by dividing space, which furtherverifies the relevant conclusions in Section 2. When com-paring the index values of the optimized points and theuniform time, it can be found that the results are close. +eoptimal distributions improve by 1.8% to 2.0% of IAE and13.1% to 17.8% of ISDE compared with the uniform timedistributions. +erefore, in the case of less precision re-quirements, the use of isochronous distribution of trajectorypoints has been able to achieve better trajectory reduction

0.35 0.4 0.45 0.5

–0.9

–0.85

–0.8

Ideal curveCurve by uniform timeCurve by Chebyshev node

–1

–0.9

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3A

ngle

s (ra

d)

0.2 0.4 0.6 0.8 10Time (s)

(a)

0.35 0.4 0.45 0.5–2.9

–2.8

–2.7

Ideal curveCurve by uniform timeCurve by Chebyshev node

0.2 0.4 0.6 0.8 10Time (s)

–3

–2.8

–2.6

–2.4

–2.2

–2

–1.8

Ang

les (

rad)

(b)

Figure 11: Joint trajectory effect based on temporal divisions. (a) HFE joint trajectory. (b) KFE joint trajectory.

Start

Particle swarm initialization

Meet the accuracy

End

Forward kinematics solving to obtain Cartesian

trajectory

Calculating boundary conditions

Joint space trajectory planning

Output result

Update particle speed and position

No

Fitness calculation

Yes

(a)

Start

Coding, initialization

Meet the accuracy

End

Forward kinematics solving to obtain Cartesian

trajectory

Calculating boundary conditions

Joint space trajectory planning

Output result

No

Fitness calculation

Yes

Selection

Crossover

Mutation

(b)

Figure 12: Intelligent optimization algorithms flowchart for point distributions. (a) PSO flowchart. (b) GA flowchart.

Complexity 11

Evaluating and updating locations by the PSO or genetic algorithm

fcn

fcn

qjy_in

t

qj posit

fcn

Trajectory_function

fcn

k positd

simin

Points_in

Motion_time

Interpolation_model Forward_kinematicsActual_position

Error_analysis

Error_accumulation

Fitness_show

Out_fitness11/sed

positd

Original_position

posit

Figure 13: Model of fitness calculation system.

Table 7: Corresponding trajectory points obtained by PSO.

Point number Time (s) X position (mm) Y position (mm)1 0 15 02 0.1024 11.3051 2.69243 0.2048 1.6519 15.71654 0.3013 2.4030 35.34045 0.4012 20.4961 54.17806 0.5027 50.8640 61.79407 0.6001 79.8367 53.98698 0.6981 97.5399 35.46849 0.7933 98.4766 16.056110 0.8969 88.7543 2.742511 1 85 0

Table 8: Corresponding trajectory points obtained by GA.

Point number Time (s) X position (mm) Y position (mm)1 0 15 02 0.1055 11.0393 2.91843 0.2031 1.7704 15.41514 0.3005 2.3279 35.16965 0.3996 20.0868 53.94256 0.5006 50.1920 61.79977 0.5999 79.7857 54.01648 0.6979 97.5207 35.51119 0.7922 98.5489 16.254110 0.8922 89.1626 3.093011 1 85 0

Table 9: Performance index values of multiple distribution types.

Distribution mode of trajectory points IAE ISDEUniform X 1.3860 1.6320Uniform arc 0.6194 0.6010Uniform time 0.2560 0.0214Chebyshev nodes 0.7830 0.3606PSO 0.2514 0.0186GA 0.2505 0.0176

12 Complexity

effect. Moreover, the computation is simple by this method.In addition, if the accuracy needs to be further improved, thePSO or GA method in here can be used to further optimizethe point distributions.

5. Simulation Analysis andPrototype Experiment

Based on the above analysis conclusions, a virtual prototypesimulation and a prototype experiment are performed tofurther verify the above optimization results.

By building a virtual prototype associated simulationplatform, the motion simulation of a virtual prototype isperformed. In this process, the input and output interfaces ofeach platform are set up for driving and feedback. +emotion time and the point distributions are inputted inSimulink to plan the joint space trajectory, and the corre-sponding joint angle output is got. +e joint angle output bySimulink is used as the input of virtual prototype for jointdriving and locomotion. +e comparison between the actualfoot-end trajectory generated by several distribution modesand the original Cartesian trajectory is shown in Figure 14.

In Figure 14, because the points obtained by PSO andgenetic algorithm are relatively close, this will causeoverlap. After considering the accuracy and convergencespeed of the two methods, the point distributions obtained

by PSO are analyzed owing to this method having a fasterconvergence speed, and the accuracy is close to geneticalgorithm. +us, the simulation of the motion process andthe trajectory curve of the foot-end are obtained, whichare shown in Figure 15.

In Figure 15, the movement process of the foot-end in atrajectory cycle is simulated. From the partially enlargedfigure of each picture and Figure 14, it can be seen that themotion trajectory of the foot-end is consistent with thedesigned trajectory curve. +en, the foot-end motioncharacteristics are analyzed, the obtained speed curve of thefoot-end is shown in Figure 16(a), and the acceleration curveis shown in Figure 16(b).

It can be seen that the foot-end velocity curve of thevirtual prototype generated by the joint trajectory plan-ning is basically consistent with the designed velocitycurve, and there is a small fluctuation around the designedspeed curve and it is relatively smooth in general com-paring Figure 16 with Figures 4 and 5. +e accelerationcurve of the foot-end is consistent with the designedacceleration curve in the overall trend, but the details aredifferent. Although the acceleration curve of the foot iscontinuous, the fluctuation range is obvious and thesmoothness is poor. It will cause the vibration and res-onance of the machine during the actual movement[44, 45]. How to optimize the boundary conditions of the

0 20 40 60 80 100 120X (mm)

Original curveUniform XUniform arc

Uniform timeChebyshev nodePSO

15 20 25 30 35

26 28 30 32 34 36 38

299300301302303304

20 22110 112 114 116

350348346344342340

334332330328326324

345

344

343

360

350

340

330

320

310

300

290Y

(mm

)

Figure 14: Comparison of different trajectory restoration effects.

Complexity 13

1.00.90.80.70.60.50.40.30.20.10.0

400.0

300.0

200.0

100.0

0.0

–100.0

–200.0

–300.0

Time (s)

Vel

ocity

(mm

/s)

X_directionY_direction

(a)

4000.0

3000.0

2000.0

1000.0

0.0

–1000.0

–2000.0

–3000.01.00.90.80.70.60.50.40.30.20.10.0

Time (s)

Acc

eler

atio

n (m

m/s

2 )

X_directionY_direction

(b)

Figure 16: Foot-end motion characteristics of virtual prototype. (a) Foot-end speed curve. (b) Foot-end acceleration curve.

(a) (b) (c) (d)

(e) (f ) (g) (h)

Figure 15: Virtual prototype motion process using the point distributions obtained by PSO algorithm. (a) Start. (b) 1/7 period. (c) 2/7period. (d) 3/7 period. (e) 4/7 period. (f ) 5/7 period. (g) 6/7 period. (h) 1 period.

Visual C++

PowerPMACIDE

ControllerPMAC CK3E-1310

Servo drivesNanotec C5-E-2-21

Quadruped robot prototype

Servo motor

Encoder Laser tracker

Figure 17: Experimental platform for bionic trajectory testing.

14 Complexity

interpolation points is the key to solve the above accel-eration unsmoothness problem, and it is also the topicthat may need to be researched after this paper.

+e experimental platform of quadruped robot proto-type is shown in Figure 17, and the interpolation pointsobtained by PSO algorithm are used as the experimentalobjects for single-leg motion experiment.

In Figure 17, the bus controller is used to control the motorthrough Ethernet communication, and the laser tracker is usedto track the motion trajectory in real time. +e actual motionprocess of the prototype is shown in Figure 18.

During the experiment, the movement was at a steadyperformance and achieved a good practical effect. How-ever, due to the gap in the keyway of the leg joint of theprototype, some structural jitter occurs when it is mag-nified to the foot-end. But, for industrial robots, the effectsmay be better due to the well assembly accuracy. +esampling effect obtained by the laser tracker is shown inFigure 19. Compared with the design trajectory in

Figure 3, it can be seen that the optimized interpolationpoints can better restore the original trajectory in theactual foot-end movement.

6. Conclusion

(1) +e Cartesian space trajectory planning combinedwith the joint space trajectory planning is imple-mented in the foot-end trajectory planning ofquadruped robots. And the piecewise quintic poly-nomials for trajectory generation are used for the twomethods, which have good coordination andconsistency.

(2) +e position distribution of interpolation pointsused in joint space trajectory is explored.+e originaltrajectory is divided into four distribution types:uniform X, uniform arc, uniform time, and Che-byshev nodes. +rough analysis and comparison, itwas found that the trajectory restoration effect basedon temporal division was better than that based onspatial division. Among the four methods, the tra-jectory reduction degree based on uniform timepartition was the highest.

(3) On the basis of the above analysis, the PSO and GAare used to optimize the distribution of interpolationpoints, and the points distribution with higher re-duction degree is obtained, which is greatly im-proved compared with the above four methods.+erefore, in the case that accuracy demand is notparticularly strict, the isochronous distributionmode can be utilized since it already has high re-duction degree and calculation convenience. Inaddition, if the accuracy needs to be further im-proved, the PSO or GA method in here can be usedto further optimize the point distributions.

(a) (b) (c) (d)

(e) (f ) (g) (h)

Figure 18: Movement process of foot-end of prototype. (a) Start. (b) 1/7 period. (c) 2/7 period. (d) 3/7 period. (e) 4/7 period. (f ) 5/7 period.(g) 6/7 period. (h) 1 period.

Figure 19: Laser tracker sampling trajectory by prototypeexperiment.

Complexity 15

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (Nos. 51965029, 61873115, and51565021) and National Key Research and DevelopmentPlan Project (No. 2017YFC1702503).

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Complexity 17