ADE690470 1 10 Research Article Advances in Mechanical Engineering 2017, Vol 9(2) 1–10 � The Author(s) 2017 DOI 10 1177/1687814017690470 journals sagepub com/home/ade An approach to the dynamic modeli[.]
Research Article An approach to the dynamic modeling and sliding mode control of the constrained robot Advances in Mechanical Engineering 2017, Vol 9(2) 1–10 Ó The Author(s) 2017 DOI: 10.1177/1687814017690470 journals.sagepub.com/home/ade Heng Shi, Yanbing Liang and Zhaohui Liu Abstract An approach to the dynamic modeling and sliding mode control of the constrained robot is proposed in this article On the basis of the Udwadia–Kalaba approach, the explicit equation of the constrained robot system is obtained first This equation is applicable to systems with either holonomic or non-holonomic constraints, as well as with either ideal or non-ideal constraint forces Second, fully considering the uncertainty of the non-ideal force, that is, the dynamic friction in the constrained robot system, the sliding mode control algorithm is put forward to trajectory tracking of the endeffector on a vertical constrained surface to obtain actual values of the unknown constraint force Moreover, model order reduction method is innovatively used in the Udwadia–Kalaba approach and sliding mode controller to reduce variables and simplify the complexity of the calculation Based on the demonstration of this novel method, a detailed robot system example is finally presented Keywords Constrained robot, Udwadia–Kalaba equation, sliding mode control, dynamic modeling, simulation Date received: 29 September 2016; accepted: January 2017 Academic Editor: Elsa de Sa Caetano Introduction A constrained robot system is a typical mechanical system The control of this kind of system usually needs some dynamic equations It is known that the robot system has characteristics of high coupling, nonlinearity, and uncertainty in trajectory tracking As a result, it is almost impossible to build a model of the robot dynamics perfectly Fortunately, this solution of this problem is vigorously worked since the constrained movement technique was proposed by Lagrange.1 He developed a Lagrange multiplier method for solving constrained movements However, in practical engineering applications, it is difficult to get the Lagrange multiplier, which leads to the difficulty of obtaining this equation Then, Gauss2 provided a new common principle for motions of constrained mechanical systems, which can be applied in constrained robot systems Gibbs3 and Appell4 also present the comprehensive equation of movement which is apprised highly by Pars;5 however, the equation is difficult to deal with large degree of freedom (DOF) Professors Udwadia and Kalaba6–9 proposed the equation of the multi-body system motion under the constraint condition, which is one of the important achievements in Lagrange mechanics field This equation is applicable to a variety of constraints, such as holonomic and non-holonomic constraints Later on, they extended their work to the non-ideal constraint system and general mechanical system The merit of Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an, China Corresponding author: Heng Shi, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China Email: shiheng@opt.ac.cn Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage) 2 their method is that the system they focus on may not meet D’Alembert’s principle,10 while other works are almost on the basis of D’Alembert’s principle Attributing to the simple and general expression, this equation has attracted more and more attentions and has been applied in many different fields In recent years, researchers have done much work to obtain the dynamic model and control of the constrained robot Liu and Liu11 got dynamic modeling of industrial robot subject to constraint using the Udwadia–Kalaba equation, and the ideal constraint force is taken into consideration Su et al.12 used a sliding mode control algorithm in the constrained robot system, but the approach ignored the constraint force which is indispensable in practical application Wang et al.13 primarily studied the variable control of nonholonomic constraints Wanichanon et al.14 proposed a general sliding control scheme on the holonomic and non-holonomic constraints On the foundation of their work, the ideal force and non-ideal force are both taken into consideration and a novel dynamic model is established Actually, a constrained robot system usually suffers from the constraint force, which is commonly caused by the end-effector of the robot being constrained on a surface In practice, the constraint force which is produced on the constraint surface cannot be ignored This constraint force can be divided into two parts One is the normal force which is regarded as the ideal constraint force Thanks to the Udwadia–Kalaba approach, it can be used to obtain the dynamic model combined with normal force in constrained robot system, and the explicit expression of the normal force can be obtained The other is the tangential force which can be seen as the non-ideal constraint force Non-ideal constraint forces often include friction force and electromagnetic force Note that the friction force cannot be ignored in the constrained robot system, but unfortunately it cannot be calculated In the simulation, the friction force can be obtained by first giving an initial condition and then introducing the sliding mode control method to track the trajectory and force The constrained robot system is a very complicated multi-input multi-output (MIMO) nonlinear system, which has several dynamic characteristics of time-varying, coupling, and nonlinear For these characteristics, neural adaptive control and sliding mode control are regularly used to control the constrained robot, and lots of researchers have achieved good results S Frikha et al.15 proposed an adaptive neural sliding mode control scheme with Lyapunov criterion for typical uncertain nonlinear systems, and neural network was used to estimate the structural model of the system H Wei et al.16 used adaptive neural network control with fullstate feedback for an uncertain constrained robot, which can effectively guarantee the performance Advances in Mechanical Engineering and improve the robustness of closed-loop system R Garcı´ a-Rodrı´ guez and V Parra-Vega17 designed a neural sliding mode control scheme for constrained robots based on Lyapunov function, which can prove the robustness of closed-loop system and finally conclude convergence of position and force tracking errors Liu et al.18 proposed neural network control which is based on adaptive learning design for nonlinear systems with state constraints, in which signals of the closed-loop system are bounded and the tracking error converges to a bounded compact set In this article, the sliding mode control19–21 is used to trajectory tracking of the end-effector on a vertical constrained surface to obtain actual values of the unknown constraint force The sliding mode control is also called the variable structure control, which is proposed by Soviet scholars Utkin and Emeleyanov The structure of the sliding mode control system is constantly changing as the current state, so that the system is moving according to a predetermined trajectory The sliding mode control method is suitable for the robot control because of its two benefits On one hand, the sliding mode control does not need the accurate mathematical model of the controlled object As mentioned above, in the constrained robot system, the non-ideal force can only be obtained by experimentation or observation Therefore, utilizing this benefit, the sliding mode control algorithm is appropriate to achieve trajectory and force tracking On the other hand, the sliding mode control is invariant to uncertainty factors such as parameters perturbation and the external disturbance This benefit can ensure the control performance of the system due to the random interference The main contributions of this article are as follows: By the Udwadia–Kalaba equation, the ideal and non-ideal forces are both taken into consideration Because the non-ideal constraint force is hardly calculated but only can be obtained by the experiment or the experience, the sliding mode control algorithm is developed for tracking the non-ideal force (the dynamic friction) in the constrained robot system In this way, the dynamic equation of the constrained robot is more complete and the non-ideal force can be obtained theoretically This contribution gives a theoretical basis for the future experiment investigation The major innovation of this article is the establishment of a new order reductive dynamic equation and sliding mode controller to describe the constrained robot motion The model order reduction method not only can simplify the complexity of the calculation but also can be extended to multi-degree of the constrained robot system Shi et al The outline of this article is organized as follows First, Udwadia–Kalaba approach is described in detail Second, the dynamic model of the constrained robot system is obtained by the order reduction and Udwadia–Kalaba approach and the sliding mode control algorithm is derived Third, the 2-DOF robot with vertical constraint is used as the example to specify and verify the correctness of the Udwadia–Kalaba approach and sliding mode control approach Fourth, some conclusions are presented Detail the Udwadia–Kalaba approach To the robot system without constraint, the dynamic equation of n-DOF robot is established with Lagrange method22 _ q_ + G(q) = t M(q, t)€q + C(q, q) _ t) + Qc (q, q, _ t) M(q, t)€q = Q(q, q, where q = ½q1 , q2 , , qn describes joint displacements t denotes applied joint torques M(q, t) is the _ symmetric and positive inertia or mass matrix C(q, q) represents coriolis and centrifugal torques G(q) denotes gravitational torques Here, assume joint displacements are independent of each other Equation (1) is written in the following form ð2Þ ð7Þ c _ t) is n-vector, which is caused by the where Q (q, q, additional constraint force and satisfies constraint conditions According to D’Alembert’s principle, constraint forces can positive, negative, or zero work under virtual displacement in the constrained system When constraint forces no work, they are called ideal constraint forces While constraint forces work, they can be named as the non-ideal constraint force which is the dynamic friction in this article Therefore, when the constrained robot system exists ideal and non-ideal con_ t) can be given by straints in the same time, the Qc (q, q, ð1Þ T _ t) M(q, t)€q = Q(q, q, constrained system, the motion equation of the constrained robot system is given by _ t) = Qcid (q, q, _ t) + Qcnid (q, q, _ t) Qc (q, q, ð8Þ _ t) denotes the ideal constraint force and where Qcid (q, q, _ t) represents the non-ideal constraint force Qcnid (q, q, Assuming that the virtual displacement24 is y, the work done by the ideal constraint force Qcid is shown as y T Qcid = ð9Þ While the work done by the non-ideal constraint force Qcnid is yT Qcnid 6¼ _ t) can be thought of the external resultant force Q(q, q, of the constraint system From equation (2), when _ and t are known, the acceleration can be q, q, obtained as follows The form of the ideal and non-ideal constraint force has been given by Udwadia and Kalaba6 _ t) _ t) = M 1 (q, t)Q(q, q, a(q, q, Qcid = M B+ (b AM 1 Q) ui (q, t) = i = 1, 2, , m1 ð4Þ and _ t) = cj (q, q, j = 1, 2, , m2 ð5Þ where u is a m1 vector and c is a m2 vector Equations (4) and (5) include all the usual varieties of holonomic and/or non-holonomic constraints Differentiating equation (4) twice with respect to time and equation (5) once with respect to time, the following matrix form23 can be obtained _ t)€q = b(q, q, _ t) A(q, q, ð3Þ It is assumed that the constraint form of this robot system can be described by m = m1 + m2 equations6 ð6Þ where A referred to as m n constraint matrix and b is a m-vector When the system is constrained and additional set of forces act on the robot system, which can be called the ð10Þ ð11Þ and 1 Qcnid = M (I B+ B)M 2 c ð12Þ where B = AM 1=2 and the superscript ‘‘ + ’’ represents the Moore–Penrose inverse matrix The vector c is a known vector, which can be obtained by experimentation or observation in a certain mechanical system.8 From above all, the equation of the constrained robot system is given by 1 M€q = Q + M B+ b AM 1 Q + M I B+ B M 2 c ð13Þ Remark Non-holonomic constraint is the constraint that contains time derivatives of the generalized coordinates of the system, which is not integrable While, non-ideal constraint is the one that does virtual work which is not equal to zero in any particle system So, non-holonomic and non-ideal constraints are Advances in Mechanical Engineering naturally different in the aspect of definition In this article, holonomic and non-holonomic constraints, as well as ideal and non-ideal constraints are used to indicate different kinds of constraints in the constrained robot systems And according to the Udwadia–Kalaba approach, the explicit equation of the constrained robot system is applicable to all holonomic and nonholonomic (ideal and non-ideal) constrained systems no matter whether they satisfy D’Alembert’s principle f(x) = where f is two times continuously differentiable Assuming that the vector x and q have such a relation x = h(q) The constrained robot is a typical mechanical system The 2-DOF robot with vertical constraints is shown in Figure below As shown in Figure 1, it is the schematic diagram of 2-DOF robot with the vertical constraint Let x = ½x1 , x2 T denotes the end-effector coordinate of the constrained robot in Cartesian coordinate system q = ½q1 , q2 T is the generalized coordinate of the system There are two perpendicular forces acting on the endeffector by the vertical plane The first force is the ideal constraint force Qcid , which is the positive pressure on the contact surface and is perpendicular to the constrained surface The second force is the non-ideal constraint force Qcnid , which remains tangent to the constrained surface Therefore, the ideal constraint force Qcid provides holding power to guarantee the endeffector for moving on the contact surface The nonideal constraint force Qcnid provides the tangential acceleration along the constrained surface The equation of the constraint is written as20 ð15Þ where h is two times continuously differentiable, then the equation of constraints in joint space is obtained F(q) = fðh(q)Þ = Model reduction ð14Þ ð16Þ In the constrained robot system, Qcid can be obtained by equation (11) Qcnid includes the dynamic friction only in the end-effector When the end-effector is moving on the vertical plane, it is the dynamic friction acting on the robot Due to the uncertainty of the Qcnid , the non-ideal constraint force is expressed as Qcnid = J T (q, t)F(t) = J T (q, t)l ð17Þ where F(t) is the dynamic friction in Cartesian space l is the associated Lagrangian multiplier J T (q, t) is the Jacobian matrix of equation (17), which can be given by J (q, t) = ∂F(q) ∂f(x) ∂x ∂f(x) ∂h(q) = = ∂q ∂x ∂q ∂x ∂q ð18Þ From equations (1), (7), (8), and (17), the dynamic equation of the 2-DOF constrained robot is given by _ t) + J T (q, t)l = t _ q_ + G(q) + Qcid (q, q, M(q, t)€q + C(q, q) ð19Þ Since the end-effector of the robot is constrained in the vertical surface, DOFs of the robot system changed from two to one Here, choose q1 as the variable to describe the motion of the constrained robot So, q2 is the remaining joint redundant variable And q2 can be donated by q1 q2 = c(q1 ) ð20Þ And then from equation (20), one can obtain # " q_ q_ = ∂c(q1 ) = L(q1 )q_ q_ = _ q_ ∂q1 q1 ð21Þ and Figure 2-DOF robot with vertical constraint _ )q_ + L(q1 )€q1 €q = L(q ð22Þ where L(q1 ) = ∂c(q1 )=∂q1 Therefore, equation (19) is expressed in the reduced form as Shi et al M1 (q1 , t)€ q1 + C1 (q1 , q_ )q_ + G1 (q1 ) + Qcid1 (q1 , q_ , t) T + J (q1 , t)l = t ð23Þ where M1 (q1 , t) = M(q, t)L(q1 ), C1 (q1 , q_ ) = M(q, t) c _ ) + C(q, q)L(q _ _ , t) = L(q ), G1 (q1 ) = G(q), Qid1 (q1 , q _ t), and J T (q1 , t) = J T (q, t) Qcid (q, q, Remark Equation (23) is the basis for the control purpose of the constrained robot system Now multiply both sides of equation (23) with LT (q1 ), the following equation is obtained LT (q1 )M1 (q1 , t)€q1 + LT (q1 )C1 (q1 , q_ )q_ + LT (q1 )G1 (q1 ) + LT (q1 )Qcid1 (q1 , q_ , t) + LT (q1 )J T (q1 , t)l = LT (q1 )t ð24Þ Equation (24) can be simplified as ML (q1 , t)€q1 + CL (q1 , q_ )q_ + GL (q1 ) + QcidL (q1 , q_ , t) = LT t ð25Þ By exploiting the structure of equations (23) and (25), three properties can be obtained:25 Property 1: Define the matrix ML (q1 , t) = LT (q1 ) M1 (q1 , t), ML (q1 , t).0 Property 2: Define the matrix CL (q1 , q_ ) = LT (q1 ) _ L (q1 , t) 2CL (q1 , q_ ) is the C1 (q1 , q_ ) and then M skew symmetric matrix Property 3: J (q1 , t)L(q1 ) = LT (q1 )J T (q1 , t) = Three properties are the basis of the design for the sliding mode control laws To obtain the practical dynamic friction in simulation, the l value can be obtained using the following three situations To be sure, the value of l can be measured by the force sensor in the practical engineering Sliding mode control for the constrained robot system In this section, a general tracking problem for the constrained robot system is considered As the desired joint position qd (t) and the desired constraint force ld are known, the desired constraint force which is known for the desired dynamic friction J T (q, t)ld is known Note that the objective of the control is to track the desired joint position and the constraint force within an acceptable error range and to satisfy the imposed constraints, Qcnidd = J T (q, t)ld and F(qd ) = A sliding mode control law is designed to make q(t) track qd (t), and Qcnid track Qcnidd as t ! ‘ Since q2 = c(q1 ), a sliding mode control law is only need to be found to satisfy q1 (t) track qd1 (t) as t ! ‘ Defining e1 = qd1 q1 ð28Þ q_ r1 = q_ d1 + Le1 ð29Þ el = ld l ð30Þ where e1 denotes the tracking error, el represents the force tracking error, qr1 is the reference trajectory, and L.0 is the tunable matrix The sliding surface is defined as s1 = q_ r1 q_ = e_ + Le1 ð31Þ sL1 = L(q1 )s1 ð32Þ The sliding controller is defined as t = M1 (q1 , t)€qr1 + C1 (q1 , q_ )q_ r1 + G1 (q1 ) + Qcid1 (q1 , q_ , t) + Kp sL1 + J T (q1 , t)lr ð33Þ Situation 1: When J (q) = 0, the value of l can be obtained by the expression of J T (q1 , t) or the definition of the dynamic friction Situation 2: When J (1) 6¼ 0, the value of l is given by where Kp The item for controlling the dynamic friction is given by t(1) M1 (1)€q1 C1 (1)q_ G1 (1) Qcid1 (1) J (1) ð26Þ where Kl From equation (34), the following equation is obtained l= Situation 3: When J (2) 6¼ 0, the value of l is given by l= t(2) M1 (2)€q1 C1 (2)q_ G1 (2) Qcid1 (2) J (2) ð27Þ lr = ld + Kl el lr l = el + Kl el = ð1 + Kl Þel ð34Þ ð35Þ Using equations (23), (31), and (33), the following equation can be obtained M1 (q1 , t)_s1 + C1 (q1 , q_ )s1 + Kp sL1 = J T (q1 , t)ðl lr Þ ð36Þ Advances in Mechanical Engineering Multiply both sides of equation (36) with LT (q1 ) and using the Property 3, equation (36) can be written as ML (q1 , t)_s1 + CL (q1 , q_ )s1 + LT (q1 )Kp sL1 = ð37Þ _ = C(q, q) p4 q_ sin q2 p4 q_ sin q2 3 G(q) = p1 e1 p4 (q_ + q_ ) sin q2 ð38Þ Differentiating equation (38) with respect to time gives _ V_ = s1 ML (q1 , t)_s1 + M L (q1 , t)s1 where e1 = g=l1 , g is the acceleration of gravity And the four unknown parameters p1 , p2 , p3 , and p4 are functions of physical parameters m1 l12 p2 = m2 l12 p3 = m2 l22 p = m2 l1 l2 p1 = ð39Þ Considering the skew symmetric property of _ L (q1 , t) 2CL (q1 , q_ ), equation (39) is expressed as M V_ = s1 ðML (q1 , t)_s1 + CL (q1 , q_ )s1 Þ ð40Þ Substituting equation (37) into equation (40), one can obtain V_ = s1 ðML (q1 , t)_s1 + CL (q1 , q_ )s1 Þ = s1 LT (q1 )Kp sL1 = sTL1 Kp sL1 = Kp s2L1 ð41Þ Since V_ is negative semi-definite and Kp is positive definite, when V_ [0, the following are obtained sL1 [0, s_ L1 [0 ð42Þ s1 [0, s_ [0 e_ [e1 [0 According to equations (35), (36), and (42), it is obvious that ð43Þ l ! ld as t ! ‘ In equation (48), because the robot has not been constrained, the vector of applied joint torque is t = ½0, 0T As shown in Figure 1, the position of the endeffector is obtained as X1 = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X2 = l12 + l22 2l1 l2 cos (2q1 ) ð49Þ where ½X1 X2 is the coordinate of the end-effector in the global coordinate system Considering the end-effector of the robot is constrained with the vertical surface, one can get l1 sin q1 + l2 sin (q1 + q2 ) = X2 ð50Þ As shown in Figure 1, the 2-DOF robot with vertical constraint is used to verify the correctness and reliability of the proposed dynamic model and the sliding mode control method Detailed matrices in equation (1) are shown in the following p + p2 + p3 + 2p4 cos q2 M(q, t) = p3 + p4 cos q2 ð48Þ l1 sin q1 €q1 l2 sin (q1 + q2 )€q1 l2 sin (q1 + q2 )€q2 = l1 cos q1 q_ + l2 cos (q1 + q2 )(q_ + q_ )2 + X€ Simulated example _ t) = C(q, q) _ q_ G(q) Q(q, q, Taking time derivation twice on equation (50), one can obtain The following can be known by LaSalle theorem e_ ! 0, ð47Þ According to equation (2), the external resultant force without constraint can be obtained as26 l1 cos q1 + l2 cos (q1 + q2 ) = X1 l lr [0 el [0 e1 ! 0, ð46Þ ML (q1 , t)s21 ð45Þ cos q1 + p2 e1 cos q1 + p4 e1 cos (q1 + q2 ) p4 e1 cos (q1 + q2 ) The Lyapunov function is taken as V= p3 + p4 cos q2 p3 ð44Þ l1 cos q1 €q1 + l2 cos (q1 + q2 )€q1 + l2 cos (q1 + q2 )€q2 = l1 sin q1 q_ + l2 sin (q1 + q2 )(q_ + q_ )2 + X€ ð51Þ Differentiating equation (49) with respect to time twice, the following equation is given by X€ = 2n 4m2 p 4n2 €q1 + q_ X€ = m3 m ð52Þ Shi et al where the three parameters m, n, and p are functions of physical parameters Two situations to get the value of l are shown as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l12 + l22 2l1 l2 cos (2q1 ) m= Situation 1: When q1 + q2 = p and q1 = 0, q = p, we get J (q, t) = From Figure 1, one can easily find that two links of the robot are placed with overlapping each other In this situation, l = is obtained, which means the constrained dynamic friction is zero Situation 2: When q1 + q2 6¼ p and J (1) 6¼ 0, J (2) 6¼ 0, l can be obtained by equations (26) and (27) ð53Þ n = l1 l2 sin (2q1 ) p = l1 l2 cos (2q1 ) Substituting equation (52) into equation (51), the second-order constraint equation23 is shown as ð54Þ _ t)€q = b(q, q, _ t) A(q, q, in which " A= " b= l1 sin q1 l2 sin (q1 + q2 ) l1 cos q1 + l2 cos (q1 + q2 ) l2 sin (q1 + q2 ) 2n m # l2 cos (q1 + q2 ) # + l2 cos (q1 + q2 )(q_ + q_ )2 l1 cos q1 q_ 21 l1 sin q1 q_ 21 + l2 sin (q1 + q2 )(q_ + q_ )2 + 4m2 p4n2 m3 q_ 21 ð55Þ Now, M, A, b, and Q have been obtained, so the ideal constraint force Qcid can be obtained according to equation (11) As shown in Figure 1, the constraint function is X1 = f(x) = ð56Þ f(q1 ) = l1 cos q1 + l2 cos (q1 + q2 ) = Since the constraint equation is F(q) = fðh(q)Þ = f(q1 ) = l1 cos q1 + l2 cos (q1 + q2 ) ð57Þ According to equation (18), the following is obtained J (q, t) = ∂F(q) = ½ l1 sin q1 l2 sin (q1 + q2 ) ∂q ð58Þ l2 sin (q1 + q2 ) Λ qd − In simulation, it is assumed that l1 = l2 = 1, m1 = m2 = According to equation (56), one can obtain The robot is constrained by the vertical surface, so 0\q1 p=2, q2 \p Then cos (q1 + q2 ) = cos (p q1 ) q d e ð60Þ The relationship between q1 and q2 can be obtained as q2 = p 2q1 ð61Þ According to equation (61), one can obtain L(q1 ) = 2 L(q1) K p JT(q1,t) s1 The sliding mode controller eλ − − Kλ Figure The control scheme of the controlled system ð62Þ For the simulation, the reduced order model is used as the controlled object The initial position is qd1 = 0:7 + 0:6 cos t and the desired dynamic friction is ld = sin t; therefore, the initial value is chosen as q = ½ 1:3 p 2:6 The controller parameters are Kp = , Kl = 8, and L = 10:0 The simulation time is set to 20 s The control scheme of the proposed control is shown in Figure e1 The calculation of the sliding mode surface ð59Þ cos q1 + cos (q1 + q2 ) = λd q1 q The constrained robot system λ q1 Figure The angle and angle speed tracking of the first link Figure The angle and angle speed tracking of the second link Figure Tracking errors of angles of the robot Simulation results are shown in Figures 3–8 Figures and show angles and angle speed tracking of the constrained robot, in which Figure Advances in Mechanical Engineering Figure Tracking errors of angle speed of the robot represents the first link and Figure represents the second link In Figure 3, solid red lines represent ideal values of q1 and dq1 and dashed blue lines represent practical values of q1 and dq1 It can be found that solid lines are almost coincident with dashed lines Similar to Figure 4, solid red lines represent ideal values of q2 and dq2 and dashed blue lines represent practical values of q2 and dq2 Due to equation 2q1 + q2 = p, as shown in Figures and 4, the relationship between q1 and q2 has been validated obviously Figures and show tracking errors of angles and angle speed of the constrained robot system It is obvious that tracking errors of angles and angle speed are almost equal to zero In the stable region of Figures and 6, the maximum error of q1 and q2 is less than 0.0005 rad and the maximum error of dq1 and dq2 is less than 0.001 rad/s The main cause of this error and stability is the switch of discontinuous features in sliding mode control, which causes the chattering of the system When the trajectory of the system gets to the switching surface, the moving point can pass through the switching surface forming the chattering Major factors in the production of chattering mainly include four aspects: time delay switch, spatial delay switch, the inertia, and discreteness of the system Figure represents the torque of the constrained robot, in which the solid red line represents torque values of the first link and the dashed blue line represents torque values of the second link Because the second link is little affected by the static moment and the first link is affected by the static and dynamic moments, t1 is greater than the t2 Figure shows the tracking and tracking error of the constrained dynamic friction, which can be seen as the non-ideal force in Udwadia and Kalaba theory It is observed that the dashed red line represents desired values of ld and the solid blue line represents practical Shi et al robot system The major innovation in this article is the establishment of a new order reductive dynamic model and sliding mode controller to describe the constrained robot motion Due to the lack of freedom, model order reduction method is creatively used in the Udwadia– Kalaba approach to complete the simulation of the constrained robot system The model order reduction method can simplify the complexity of the calculation and can be extended to multi-degree of constrained robot system A simple 2-DOF robot system with the vertical constraint is used to illustrate the methodology proposed in the article From several simulation results, it can be found that the tracking errors are almost equal to zero So, the proposed model and control method are feasible, correct, and valid Figure The torque of the constrained robot Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article Funding The author(s) received no financial support for the research, authorship, and/or publication of this article References Figure The tracking and tracking error of the dynamic friction values of l It can be found that the tracking error between l and ld is almost equal to zero From the above figures, these simulation results show that the dynamic model and the sliding mode control algorithm are achieved successfully Conclusion In this article, a novel approach to the dynamic modeling and sliding mode control of the constrained robot system is proposed By the Udwadia–Kalaba equation, expressions of the ideal and non-ideal forces are obtained and then the dynamic equation of the constrained robot system is established Because the nonideal constraint force is hardly calculated, the sliding mode control algorithm is presented for tracking the non-ideal 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(23) with LT (q1 ), the following equation is obtained LT (q1 )M1 (q1 , t)€q1 + LT (q1 )C1 (q1 , q_ )q_ + LT (q1 )G1 (q1 ) + LT (q1 )Qcid1 (q1 , q_ , t) + LT (q1 )J T (q1 , t)l = LT (q1 )t ð24Þ Equation... M1 (q1 , t)€ q1 + C1 (q1 , q_ )q_ + G1 (q1 ) + Qcid1 (q1 , q_ , t) T + J (q1 , t)l = t ð23Þ where M1 (q1 , t) = M(q, t)L(q1 ), C1 (q1 , q_ ) = M(q, t) c _ ) + C(q, q)L(q _ _ , t) = L(q ), G1... b= l1 sin q1 l2 sin (q1 + q2 ) l1 cos q1 + l2 cos (q1 + q2 ) l2 sin (q1 + q2 ) 2n m # l2 cos (q1 + q2 ) # + l2 cos (q1 + q2 )(q_ + q_ )2 l1 cos q1 q_ 21 l1 sin q1 q_ 21 + l2 sin (q1 + q2