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Hindawi Journal of Robotics Volume 2021, Article ID 9913335, 11 pages https://doi.org/10.1155/2021/9913335 Research Article Online Dynamic Tip-Over Analysis for a Wheeled Mobile Dual-Arm Robot with an Improved Tip-Over Moment Stability Criterion 1 1 2 2,3 Xianhua Li , Liang Wu , Qing Sun , and Tao Song School of Mechanical Engineering, Anhui University of Science and Technology, Huainan 232001, China School of Mechatronics Engineering and Automation, Shanghai University, Shanghai 200444, China Shanghai Robot Industry Technology Research Institute Co., Ltd., Shanghai 200063, China Correspondence should be addressed to Xianhua Li; xhli01@163.com Received 29 March 2021; Revised 6 June 2021; Accepted 25 June 2021; Published 12 July 2021 Academic Editor: Weitian Wang Copyright © 2021 Xianhua Li 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. Tip-overstabilityanalysisiscriticalforthesuccessofmobilemanipulationofthedualarm,especiallyinthecasesthatthedualarm or the mobile platform moves rapidly. Due to strong dynamics coupling between the dual arm and mobile platform, online evaluation of dynamic stability of the mobile dual-arm robot still remains challenging. *is paper presents an improved tip-over moment stability criterion dealing with the dual arm and mobile platform interaction and proposes an algorithm for calculating the tip-over stability margin of the arm end in the workspace to analyze the dynamic stability of the wheeled mobile dual-arm robot. *e simulations on a four-wheeled mobile dual-arm robot validate the correctness and feasibility of the proposed method. for the obstacle environment [8]. Kagami et al. [9] described 1. Introduction afastdynamicallyequilibratedtrajectorygenerationmethod *e wheeled mobile dual-arm robot (WMDAR) is an for a humanoid robot based on the relationship between the emerging class of robots which have capabilities of both robot’scenterofgravityandtheZMP.However,ifthecenter moving and manipulation. *e WMDAR is usually human- of mass of the robot system changes, ZMP is not sensitive to robot collaboration, enabling it to be widely employed in the stabilityof thesystem. *erefore,Papadopoulos andRey home service, restaurant service, and medical treatment deﬁned the tipping stability margin according to the force- [1–3]. However, the WMDAR is a kind of system with angle (FA) margin criterion and described a real-time unstable structure, which may tip-over under the action of rollover prediction and prevention scheme based on static diﬀerent dynamic factors or external interference, especially and dynamic force-angle measure [10, 11]. *is criterion for the mobile service robot with small mobile platform, ignores the reaction force and moment of the manipulator acting on the moving platform. *en, Moosavian and Ali- large variation of system mass distribution, and bearing extraload in work. pour proposed a moment-height stability (MHS) measure Currently, some tip-over stability criteria have been for the wheeled mobile robot considering robot dynamics applied in mobile robots [4–9]. Among the many stability and system gravity center [12–15]. In addition, some other criteria, the zero moment point (ZMP) is the most popular. stability criteria were applied to the stability detection of the Sugano et al. presented concepts of the stability degree and mobile manipulator. Ghassempoor and Sepehri proposed a the valid stable region based on the zero moment point methodtomeasurethestabilityaccordingtotheenergylevel (ZMP) [7]. Korayem et al. proposed an algorithm for de- of the moment acting on the support boundary [16]. A termining the maximum load carrying capacity of a mobile method was presented for tip-over stability analysis of a manipulatorconsideringtip-overstabilitybasedontheZMP wheeled mobile manipulator based on tip-over moment by 2 Journal of Robotics Guo et al. [17] *e normal bearing force criterion [18] re- 2. Kinematic and Force Model of the Wheeled quires force sensors to measure the bearing force, which is Mobile Dual-Arm Robot costly. At present, many works have been performed on the tip-over stability criterion of the mobile robot, but, at the 2.1.KinematicModeloftheWheeledMobileDual-ArmRobot. same time, some papers focus on the mobile robot tip-over *eWMDARconsistsofafour-wheelmobileplatform,awaist, avoidance algorithm. and the dual arm mounted on the mobile platform, depicted in Moubarak and Ben-Tzvi [19] proposed a global optimal Figure 1. *e four-wheel mobile platform is composed of a attitude convergence algorithm for redundant serial robots, platform, two driving wheels, and two driven wheels, in which whichcanpreventtippingwithoutconsideringtheinﬂuence twodrivingwheelsgoforwardorturnthroughdiﬀerentialdrive. of joint velocity and acceleration on tipping stability. Rey TodescribethemotionoftheWMDAR,coordinatesystems and Papadopoulos [11] used the FA measure method for wereestablished,i.e.,theworldframe O X Y Z ,therobot W W W W initial conﬁguration of the robot to avoid the robot tipping bodyframe O X Y Z ,theleftarmframe O X Y Z ,theend S S S S L L L L over. Based on an adaptive neural fuzzy algorithm, Li and frame of left arm O X Y Z , the right arm O X Y Z , LL LL LL LL R R R R Liu [20] utilized self-motions of redundant mobile ma- and the end frame of right arm O X Y Z , as shown in RR RR RR RR nipulators to improve a robot’s stability. Ding et al. [21] Figure 2(a). And, m is the mass of the mobile platform and the proposed a real-time tipping avoidance algorithm to reduce body, and O is the center of mass of the mobile platform and the transmission of tipping torque by adjusting the ma- the body. In the four-wheel mobile platform, points p , p , p , 1 2 3 nipulator posture or changing the robot speed, which can and p are the contact points between the mobile platform and eﬀectively avoid the robot tipping. Many environments and thegroundandthefourblacksolidlinesconnectingtheadjacent scenarios contain rough and irregular terrain and are dif- two points are the four tip-over axes of the robot system, as ﬁcult for robots. Agheli and Nestinger [22] presented a showninFigure2(b).Meanwhile,therelevantparametersofthe multilegged reactive stability control method for main- robotsystemare showninTable1, andthe positionparameters taining system stability under external perturbations. Feng ofthepointinthetableareexpressedinthereferencecoordinate et al. [23] introduced a new method for evaluating the frame O X Y Z . S S S S stability of robots in rugged terrain and proposed an al- *en,thetransformationmatricesbetweendiﬀerentframes gorithm for automatically realizing self-balancing of robots. aregivenasfollows,andtheposeoftheendofthearmcouldbe Kashyap and Parhi [24] utilized the LIPM plus ﬂywheel obtained by the screw theory [25], as shown in equation (3): model (LIPPFM) for analysis of the complete dynamic cos θ −sin θ 0 d motion of the humanoid robot. It can be seen that the m m xm ⎢ ⎤ ⎥ ⎡ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ previous research on the stability maintenance of the robot ⎢ sin θ cos θ 0 d ⎥ ⎢ ⎥ W ⎢ m m ym ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ � , (1) T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ S ⎢ ⎥ motionprocessmainlyfocusedontheattitudechangeofthe ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 1 0 ⎥ ⎣ ⎦ manipulator of the single-arm mobile robot through the 0 0 0 1 analysis of the above literature. However, there is no re- search on the attitude change of the two arms and the cos θ 0 sin θ 0 l l overturning compensation caused by the speed and accel- ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ eration of the two arms. ⎢ L ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 1 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ In this paper, an improved tip-over moment stability S ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ T �⎢ ⎥, ⎢ ⎥ ⎢ ⎥ L ⎢ ⎥ ⎢ ⎥ criterion is proposed dealing with dual-arm and mobile ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −sin θ 0 cos θ d ⎥ ⎢ ⎥ ⎢ l l ⎥ ⎢ ⎥ platform interaction. Meanwhile, an algorithm for calcu- ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ lating the tip-over stability margin of manipulator work- 0 0 0 1 space is also presented to analyze the dynamic stability of (2) cos θ 0 sin θ 0 l l WMDAR. *is paper proposes a method to study the sta- ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ bility of WMDAR, and this algorithm is very important for ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ L ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ the follow-up research. *e dynamic stability of the robot ⎢ 0 1 0 − ⎥ ⎢ ⎥ ⎢ ⎥ S ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ T �⎢ ⎥, ⎢ ⎥ ⎢ ⎥ can be studied by integrating the algorithm into the control r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ elements,whichlaysafoundationforthetrajectoryplanning ⎢ −sin θ 0 cos θ d ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ l l ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ of the robot to tip-over avoidance. 0 0 0 1 *is paper is divided into six sections. *e kinematics and force model of this robot are analyzed in Section 2. An n o a p improvedtip-overmomentstabilitycriterionispresentedin x x x x ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Section 3. *en, in Section 4, the workspace pose dis- ⎢ ⎥ ⎢ n o a p ⎥ ⎢ ⎥ ⎢ ⎥ ξ θ ξ θ ⎢ y y y y ⎥ ⎢ ⎥ 1 1 6 6 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ g (θ) � e , . . . , e g (0) � ⎢ ⎥. (3) ⎢ ⎥ cretizationissolvedandanalgorithmforcalculatingthetip- st st ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ n o a p ⎥ ⎢ ⎥ ⎣ ⎦ z z z z over stability margin of the manipulator is proposed. In 0 0 0 1 Section 5, simulation in MATLAB software is carried out in order to validate correctness of this improved tip-over Forthesakeofsimpliﬁcationofanalysisandcomputations stability criterion and an algorithm for calculating the tip- conducted in the paper, the following assumptions are made: over stability margin of workspace to analyze the stability of WMDARisproposed,andinSection6,someconclusionson (1) *e ground is even, and no surface shrinkage is them are drawn. considered Journal of Robotics 3 Figure 1: Model of the wheeled mobile dual-arm robot. Z p r 1 O O R L r θ X L p p 4 2 g Z RR LL RR O LL RR LL X LL RR m p xm Y d W ym (a) (b) Figure 2: Coordinates’ system deﬁnition. (a) Dual-arm robot system. (b) Platform system. Table 1: Related parameters of the robot system. the arm end to joint 1 can be obtained through the iterative relationship between the links based on the Newton Euler Parameters Values method and screw theory. p (m) [0.604, 0, 0] p (m) [0, 0.604, 0] p (m) [−0.604, 0, 0] 2.2.1. .e Speed of Link. *e Jacobian matrix of each link p (m) [0,−0.604, 0] can be obtained by combining the Newton Euler method O (m) [0,−0.399, 1.278] and screw theory, as shown in the following equation: O (m) [0, 0.399, 1.278] O (m) [0, 0, 0.0658] c J � Ad ^ ξ , . . . , Ad ^ ξ , . . . ,0 , i � 1,2, . . . , n. (4) ξ θ ξ θ i 1 1 1 i i i e e m(kg) 60 *erefore, the relationship between the speed of each (2) All wheels are always in point contact with the joint and the speed of the ith link was ground, i.e., no slippage of the wheels occurs V � J θ. (5) i i (3) *e dual arm and body are rigidly connected with the platform, and the links and joints of the ma- nipulator are rigid 2.2.2. .e Acceleration of Link. *e acceleration of each link (4) *e mobile platform and the body were taken as a can be calculated by deriving the following equation: whole because the inﬂuence of the robot’s body € _ _ _ (6) V � J θ + J θ, i i motion is not considered in this paper. where J � [V × Ad ξ , . . . , V × Ad ξ , . . . ,0], i � i 0 1 i−1 i 2.2. Force Model of the Wheeled Mobile Dual-Arm Robot. ξ θ ξ θ 1 1 i i e e In this paper, the dynamic model was mainly aimed at the 1,2, . . . , n, where V is the velocity of the base (body and dynamic model of the manipulator. *e force/moment of mobile platform) in equation (5). d 4 Journal of Robotics 2.2.3. Force/Moment Equation of Link. *e force balance f f z2 z1 equation of ith link can be obtained, as shown in the fol- lowing equation: z2 z1 y2 y1 J J G E ∗ O L1 _ R1 (7) w � w − w + w + w � I V + V × I V, i i i i i i+1 i i m m y2 y1 x1 where w isthegeneralizedjointforce/momentproducedby i x2 x1 the ith joint, w is the generalized joint force/moment f x2 i+1 produced by the joint i +1 exerted on link i, w is the force/ moment of gravity exerted on link i, w is the sum of other ∗ ∗ external forces/moment, I � Ad I (Ad ) , I is the i i0 i ξ θ ξ θ i i i i e e space inertia of the current manipulator conﬁguration, and I is the spatial inertia of the link i in the initial conﬁgu- i0 p i+1 Mobile platform ration of the manipulator. *e above formula was derived from the arm base coordinate frame (for example, coordi- nate frame O X Y Z for left arm). *e force/moment l L L L L i balanceequationofjoint icanbeobtainedbycombiningthe S Newton Euler method and screw theory, as shown in the Z p W Y i following equation: J J G E ∗ Tip-over axis (8) w � w − w − w + I V + V × I V. i i i i i+1 i i W Equation (8) provides a reverse iterative method to calculate the joint constraint force/moment, which can be W calculated from the end eﬀector to the last joint n of the arm Figure 3: Force and moment model of the WMDAR. untiljoint1.*econstraintforce/momentonjoint1andthe force/moment of the arm acting on the body and mobile 2.2, and the force/moment consists of components in three platform are reciprocal from Figure 3. directions, as shown in the following equation: 3. Improved Tip-Over Moment T T (9) −w � f , m , M M Stability Criterion In this section, we have derived a new tip-over moment ⎧ ⎪ f f f f � , 1x 1y 1z stability criterion for WMDAR with consideration of dual (10) m � m m m . arm-mobile platform interactions. Figure 3 depicts the 1x 1y 1z variousforcesandmomentexertingonthebodyandmobile *etip-overmoment(TOM)onthetip-overaxis(TOA) platform. *e reaction wrench from the arm onto the body a can be calculated through the above calculation: ii+1 and mobile platform is expressed as −w based on Section TOM � f × l · a + m · a + f × d · a (k � 1and2). (11) i g i ii+1 M ii+1 M i ii+1 j�1 *e ﬁrst item in equation (11) is the moment of gravity the coordinates of two adjacent wheel-terrain contact points n n exerting on the TOA of the body and mobile platform. *e ( p , p ), i.e., i i+1 n n p − p second item is the moment of the force and moment pro- i+1 i a � . (12) ii+1 n n duced by both arms exerting on the TOA. In this paper, the p − p i+1 i robot system was divided into three modules, the body and According to the dynamic method of the rigid body mobile platform, the left arm, and the right arm. *erefore, translation, themomentsofboth armsandbodyandmobile f denotes thegravity of the body andmobile platform.*e platform relative to the TOA a were calculated. For the gravity of the body and mobile platform and the force/ ii+1 WMDAR, the minimum TOM exerting on TOA of the moment of the left or right arm exerting on the TOA for the mobile platform is as follows: robot system play an important role in the stability of the system. TOM � minTOM ,TOM , . . . ,TOM . (13) 1 2 n If tipping occurs, the robot will tip over outward along the TOA formed by two adjacent wheels, where a rep- Equation(13)indicatesthatwhentheminimumTOMof ii+1 resents a unit vector for the TOA, which can be obtained by the robot system is less than 0, that is, the TOM of the robot Journal of Robotics 5 system along the TOA is outward, the robot system will tip over. *erefore, the tip-over stability margin (TOSM) is deﬁned as TOM φ � , (14) TOM norm φ � minφ ,φ , . . . ,φ , (15) 1 2 n where TOM represents a constant value, which is the norm minimumTOMexertingonTOAwhentheWMDARisina steady state, that is, TOM >0. *erefore, the stability of norm the WMDAR system can be determined by the TOSM. When φ>0, that is, the minimum TOM on the TOA will be greater than 0, and the WMDAR system will be stable; Figure 4: Discrete workspace of the manipulator. however, when φ<0, it means that the momenton the TOA is outward, which means that the system may tip over. 4.2. Pose Discretization of the Arm Workspace. *ere are many possibilities for the pose of the end at this distribution 4. Calculating TOSM in the Arm Workspace point, which will aﬀect the conﬁguration of the arm. *erefore, the pose of the end was discretized to study the 4.1. Position Discretization of Arm Workspace. In order to tip-overstabilityofWMDAR,inwhichthearmhasdiﬀerent studytheinﬂuenceofarmmotiononthetip-overstabilityof conﬁguration. Firstly, a sphere with radius of 1 was estab- WMDAR,theworkspaceofthearmshouldbediscretized.In lished on the workspace point, and the center of the sphere this paper, we used the following method to solve the arm coincides with the workspace point. A spherical coordinate workspacediscretizationandchosetheleftarmtointroduce system with the center of the sphere as the origin was the position and posture discrete method. Firstly, a sphere establishedonthesphere.*en,thepositionofeachuniform whose radius is the total length L of the arm was established point on the surface of the sphere relative to the spherical based on the coordinate system (O X Y Z ). *en, the L L L L coordinate system was solved by using the uniform distri- radius of the sphere was divided into N parts, and all butionalgorithm[26],asshowninFigure5(a).*epositions spheres with radius from the coordinate origin to each bi- of these uniform points can be calculated by the following section point were established. *e radius calculation for- formula: mula of all spheres is as follows: θ � arccos h , k k kL r � k � 1,2, . . . , N . (16) s r 2(k − 1) h � −1 + 1≤ k≤ N , k p N − 1 *e distribution points on the surface of each sphere p were taken, as shown in Figure 4, and all the distribution points can constitute the discrete workspace points of the 3.6 1 ⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ����� ⎠ β � β + (mod(2π)) 2≤ k≤ N −1. arm. k k−1 1 − h Assuming that the radius of the sphere is r , the position k of distribution points can be obtained by the following (21) formula based on the coordinate system (O X Y Z ): L L L L When k � 1 or k � N , β � β � 0: p 1 N 2i − N − 2 π (17) β θ � , ⎧ ⎪ N ⎪ x � sin β cos , ⎪ k k 2i − 1 − N π b ⎪ ϕ � , (18) 2 N −1 y � sin β sin , b ⎪ k k ⎪ (22) x � r cos θ cos ϕ , ⎧ ⎪ ⎪ s i j ⎪ ⎪ ⎪ β ⎪ k y � r sin θ cos ϕ , (19) ⎩ z � cos , s i j k z � r sin θ , s i p � x y z . k k k k p � [x, y, z] , (20) i,j,s *en, we take the direction from the sphere center to where N and N represent the step size of θ and ϕ, re- each uniform point as the z-axis direction of each discrete a b spectively, that is to say, they indicate the density of dis- pose coordinate system, and the direction of the x-axis can tribution points on the surface of the sphere. be set arbitrarily, as shown in Figure 5(b). *erefore, the 6 Journal of Robotics i � 1,2, . . . , N , ψ � min φ , (26) i,j j � 1,2, . . . ,8. *e TOSM of each workspace point represents the tip- over stability of the system when the end of the arm is located at the point. When ψ >0, it indicates that the system is always in a stable state, and the larger the value, the better the stability of the system; otherwise, when ψ <0, the system may tip over. (a) (b) 5. Motion Analysis of Tip-Over Stability of Figure 5: Uniform points on the discrete spheres and their Z-axis WMDAR System distribution. We proposed a modular decomposition method in order to verifythecorrectnessofalgorithmcalculatedTOSM.Firstly, pose matrix of all the discrete pose coordinate systems the body and mobile platform, left arm, and right arm of the relative to the sphere coordinate system on a workspace robot system weredivided into three modules.In thispaper, point can be expressed as follows: we keep the right-arm module as the initial state in the c 0 s c −s 0 θ θ β β k k k k simulation calculation, as shown in Figure 2(a), and the ⎢ ⎥ ⎡ ⎢ ⎤ ⎥⎡ ⎢ ⎤ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ 0 1 0 inﬂuenceoftherightarmonthestabilityoftherobotsystem R � R R � ⎢ s c 0 ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ k z,β y′ ,θ ⎢ β β ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ k k ⎣ k k ⎦⎢ ⎥ ⎣ ⎦ is only the gravity. At the same time, we control the 0 0 1 −s 0 c θ θ k k movementoftheleft-armmoduleandstudytheinﬂuenceof (23) c c −s c s arm motion on the stability of the robot system. β θ β β θ k k k k k ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ s c c s s ⎥ � ⎥, ⎢ β θ β β θ ⎥ ⎢ k k k k k ⎥ ⎢ ⎥ ⎣ ⎦ −s 0 c 5.1. Static Case. In this section, the tip-over stability of the θ θ k k robot system was studied when the left arm was in a static where R denotes the matrix after rotating β around z- state in its workspace. *e discrete parameters of workspace z,β k axis, R denotes the pose matrix after rotating θ around points were selected as follows: N � 20, N � 31, and y′ ,θ k k r a y-axis, and s and c are the abbreviations of sin and cos, N � 31. *e joint velocity and acceleration of the arm were respectively. both 0, and the end load of the arm was 5kg. *e number of *e pose matrix of each discrete pose coordinate system uniform discrete pose in the workspace point was taken as to the sphere coordinate system is shown as follows: follows: N � 40. *en, the TOSM of the robot system at each workspace point was solved through combining with R 0 T � . (24) the relevant parameters of the robot, and the TOSM of each 0 1 workspace point in its workspace is drawn, as shown in Figure 6. *e pose matrix of all the points in the workspace rel- Each ball represents a workspace point of the arm, and ative to the base coordinate system (O X Y Z ) of the arm L L L L the color of the ball reﬂects the size of the TOSM at this through equation (20) can be obtained: point. *e maximum tip-over stability margin is 0.91, the R p minimum is 0.69, and the average is 0.80 from the simu- k i,j,s T � . (25) i,j,s,k lation results. In addition, it can be seen that the shape of 0 1 the discrete space is not a sphere, but a ﬂat shape in the y- axis direction, which is due to the particularity of the modular manipulator conﬁguration. *ere is little diﬀer- 4.3. Calculation of TOSM of the Arm Workspace. *e ence in the ψ value in the static workspace from Figure 6. discrete pose of a point in the workspace can be obtained by When the position of the arm end is farther away from the equation(25).*einversekinematicsanalysisshowsthatthe origin of the base coordinate, the value of ψ becomes posewillcorrespondtoeightgroupsofconﬁgurationsofthe smallerandsmaller,thatis,thestabilityoftherobotsystem armwhentheendofthearmisinthispose(iftheendcannot is worse. From the sectional view, it can be seen that the reach the point, there is no solution). *e joint angle of each value of ψ close to the origin of the coordinate system is group arm conﬁguration can be obtained according to the larger, that is, the better the stability of the robot. inverse kinematicsanalysis, andtheforce/momentof joint 1 can be obtained by arm dynamics. *en, the TOSM of each pose can be calculated based on Section 3. *e TOSM in the 5.2. Joint Speeds’ Case. *e purpose of considering joint workspace point of the arm was deﬁned as the minimum speeds is to investigate the eﬀect of the coupling term due to TOSM of all conﬁgurations corresponding to all discrete centrifugal forces and gyroscopic moments on tip-over posesontheworkspacepointwastakenastheTOSMofthis stability.Inthissection,alljointaccelerationsaresettozero. point: *ere are many kinds of joint speeds of the arm in any Y (m) Y (m) Journal of Robotics 7 1.00 1.00 1.00 1.00 0.80 0.80 0.75 0.75 0.50 0.50 0.25 0.25 0.00 0.00 0.60 0.60 –0.25 –0.25 –0.50 –0.50 –0.75 –0.75 –1.00 –1.00 –1.0 –1.0 –1.0 0.40 –1.0 0.40 –0.5 –0.5 –0.5 –0.5 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 0.20 0.20 1.0 1.0 (a) (b) 1.0 1.00 1.00 1.00 0.75 0.50 0.80 0.5 0.80 0.25 0.00 0.60 0.0 0.60 –0.25 0.40 –0.50 0.40 –0.5 –0.75 0.20 –1.00 0.20 –1.0 –1.0 –0.5 0.0 0.5 1.0 1.0 0.5 0.0 –0.5 –1.0 X (m) X (m) (c) (d) Figure 6: TOSM of the system in the workspace under the static condition. (a) Axonometric view. (b) Section view. (c) X-Y plane. (d) X-Z plane. Table 2: Diﬀerent conﬁgurations of the manipulator. possible conﬁguration of the arm. In order to ﬁnd out the situation that has a great impact on the tip-over stability of Arm conﬁguration Joint angle (rad) the robot system, the conﬁguration of the arm was calcu- 1 [π/2, π/2, π/2, 0, 0, 0] lated, as shown in Table 2. *e speed of the ﬁrst three joints 2 [π/3, π/3, π/3, 0, 0, 0] wassettochangefrom −π/2to π/2,atthesametime,andthe 3 [π/6, π/6, π/6, 0, 0, 0] load at the end of the arm was 5kg, and then, the change of system TOSM with angular velocity can be obtained, as shown in Figure 7. obtained, and the TOSM of arm workspace was drawn, as It can be seen from Figure 7 that the TOSM values in the shown in Figure 8. forward and reverse directions are symmetrical under the *e maximum TOSM is 0.83 and the minimum is 0.22 threeconﬁgurations,andwhenthejointspeedis0,theTOSM from Figure 8. *e TOSM in the whole workspace is greater values of the three conﬁgurations are the largest, that is, the than 0, which means that the robot is always in a stable state. systemstabilityisthebest.Whentheﬁrstthreejointspeedsof *ediﬀerencebetweenthemaximumvalueandtheminimum thearmareatthemaximumvalueintheforwarddirectionor value is 0.61 from the color distribution. Moreover, when the themaximumvalueinthereversedirection,theTOSMofthe positionisfartherawayfromtheoriginofthebasecoordinates robotsystemistheminimum,thatis,thestabilityoftherobot on the X-Y plane, the TOSM tends to be smaller, that is, the system is the worst. *erefore, the maximum value of joint stability of the robot is the worse. It can be seen from the speed was considered, that is, the case of the ﬁrst three joints sectional view that the TOSM in the area close to the origin is with π/2rad/s. relativelylarge,thatis,thestabilityoftherobotsystemisbetter. *ejointspeedoftheﬁrstthreejointsofthemanipulator _ _ _ was set as θ � π/2rad/s, θ � π/2rad/s, and θ � π/2rad/s, 1 2 3 while the joint speed of the last three joints was 0. *en, the 5.3. Joint Accelerations’ Case. *e purpose of considering robot system TOSM of each workspace point can be joint accelerations is to investigate the eﬀect of inertia forces X (m) X (m) Y (m) Z (m) Z (m) Z (m) Y (m) Y (m) 8 Journal of Robotics 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 –90 –60 –30 0 30 60 90 Joint speeds Conﬁguration 1 Conﬁguration 2 Conﬁguration 3 Figure 7: Variation of tip-over stability margin with joint speed in diﬀerent conﬁgurations. 1.00 1.00 1.00 1.00 0.80 0.80 0.75 0.75 0.50 0.50 0.25 0.25 0.00 0.00 0.60 0.60 –0.25 –0.25 –0.50 –0.50 –0.75 –0.75 –1.00 –1.00 –1.0 –1.0 0.40 –1.0 –1.0 0.40 –0.5 –0.5 –0.5 –0.5 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 0.20 0.20 1.0 1.0 (a) (b) 1.00 1.0 1.00 1.00 0.75 0.5 0.80 0.50 0.80 0.25 0.0 0.60 0.00 0.60 –0.25 0.40 –0.50 0.40 –0.5 –0.75 0.20 0.20 –1.0 –1.00 –1.0 –0.5 0.0 0.5 1.0 1.0 0.5 0.0 –0.5 –1.0 X (m) X (m) (c) (d) Figure 8: TOSM of the system in the workspace under the speed condition. (a) Axonometric view. (b) Section view. (c) X-Y plane. (d) X-Z plane. X (m) X (m) Y (m) Z (m) TOSM value Z (m) Z (m) Y (m) Y (m) Journal of Robotics 9 1.0 0.95 0.9 0.85 0.8 0.75 0.7 0.65 –90 –60 –30 0 30 60 90 Joint accelerations Conﬁguration 1 Conﬁguration 2 Conﬁguration 3 Figure 9: Variation of TOSM with joint acceleration in diﬀerent conﬁgurations. 1.00 1.00 1.00 0.75 0.80 0.80 0.75 0.50 0.50 0.25 0.25 0.00 0.00 0.60 0.60 –0.25 –0.25 –0.50 –0.50 –0.75 –0.75 –1.00 –1.00 –1.0 0.40 –1.0 0.40 –1.0 –0.5 –0.5 –0.5 –0.5 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 0.20 0.20 1.0 1.0 (a) (b) 1.0 1.00 1.00 1.00 0.75 0.5 0.50 0.80 0.80 0.25 0.0 0.60 0.00 0.60 –0.25 –0.50 0.40 –0.5 0.40 –0.75 –1.0 –1.00 0.20 0.20 –1.0 –0.5 0.0 0.5 1.0 1.0 0.5 0.0 –0.5 –1.0 X (m) X (m) (c) (d) Figure 10: TOSM of the system in the workspace under the acceleration condition. (a) Axonometric view. (b) Section view. (c) X-Y plane. (d) X-Z plane. X (m) X (m) Y (m) Z (m) TOSM value Z (m) Z (m) 10 Journal of Robotics Table 3: Comparsion of three cases. Situation Maximum value Minimum value Average value *e diﬀerence between the two Static 0.91 0.69 0.80 0.22 Speed 0.83 0.22 0.56 0.61 Acceleration 0.89 0.60 0.78 0.29 and moments on tip-over stability. In this section, all joint given range of static loads, joint speeds, and accelerations. speeds are set to zero as well. *ere are many kinds of *is paper proposes a method to study the stability of accelerations of each joint in any possible conﬁguration of WMDAR, and this algorithm is very important for the the manipulator. In order to ﬁnd out the situation that has a follow-upresearch.*edynamicstabilityoftherobotcanbe greatimpactonthetip-overstabilityoftherobotsystem,the calculated by integrating the algorithm into the control angular accelerations of the ﬁrst three joints were set to elements, which lay a foundation for the trajectory planning 2 2 change from −π/2rad/s to π/2rad/s at the same time; the of the robot to tip-over avoidance. three conﬁgurations of the arm are shown in Table 2. *en, thechangeofTOSMwithjointaccelerationcanbeobtained, Data Availability as shown in Figure 9. *e data used to support the ﬁndings of the study are in- It can be seen from Figure 9 that the TOSM of con- cluded within the paper. ﬁguration 1 is symmetrically distributed in the forward and reverse directions, and the minimum value is obtained at 2 2 −π/2rad/s orπ/2rad/s ,andthestabilityofthesystemisthe Conflicts of Interest worst. However, conﬁguration 2 and conﬁguration 3 *e authors declare that they have no conﬂicts of interest. gradually increase with joint acceleration, and the worst stability of the system is at −π/2rad/s . *erefore, the ac- Acknowledgments celerations of ﬁrst three joints were set as θ � −π/2rad/s, € € θ � −π/2rad/s, and θ � −π/2rad/s, while the joint accel- 2 3 *is work was supported by the Opening Project of eration of the last three joints was 0. *en, the robot system Shanghai Robot R&D and Transformation Functional TOSM of each workspace point can be obtained, and the Platform (K2020468), Graduate Core (First-Class) Curric- TOSM was drawn, as shown in Figure 10. ulum Construction Project of Anhui University of Science *eTOSMinthewholeworkspaceisgreaterthan0from andTechnology(2020HX010),andNationalNaturalScience Figure 10, which means that the robot system is always in a FoundationofChina(61803251)andinpartbytheGraduate stable state. *e maximum and minimum values of TOSM Innovation Fund of Anhui University of Science and are 0.89 and 0.60, respectively. *e TOSM in the negative Technology (2020CX2042). direction of the X-axis is smaller than that in the positive direction of X-axis from the color distribution, so the sta- References bility of the system is worse. 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Journal of Robotics – Hindawi Publishing Corporation
Published: Jul 12, 2021
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