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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;
[email protected]
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:[email protected]://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
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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
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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
