Keywords

1 Introduction

The study of robot and robot dynamics has been carried out over a decades using different types of approaches. One of the methods is to reduce the order of the multi-robot system with some of the properties of the model [1]. Deshpande et al. [2] compare computational complexities of various forms of the equations of motion for open chain systems for dynamics of robot manipulator. Yamane and Nakamura [3] describe a simulation model of human body for its motion analysis, and the algorithms were developed in robotics which are combined with physiological models to solve the difficulties arise of handling real human structure.

Dynamic modeling and simulation of the industrial robot, Stiiubli TX40, are presented by Cheraghpour et al. where a precise simulation for experimental analysis of kinematics, dynamics, and control is described [4]. In this study, inertial and geometric parameters are accurately measured and recorded in the software database. Dynamic modeling of a nonholonomic wheeled manipulator is carried out which consists of elastic joints and a self-directed wheeled movable platform [5, 6]. Gibbs–Appell (G–A) recursive algorithm is adopted to avoid computation of Lagrange multipliers. The proposed algorithm recursively and methodically derives the dynamic equation, moreover all mathematical processes are done by 3 × 1 and 3 × 3 matrices to improve the computational complexity. Also, local coordinate system is assigned to all dynamic equations of a link by which the algorithm was generalized where the implementation and simulation of a greater degrees of freedom wheeled movable robotic manipulator become easier. However, the damping and friction were not considered to improve joint modeling and flexibility. Lagrange–Euler dynamics are presented in [7,8,9] where the aim is to derive simple and well-structured dynamic equations of motion. Again, the Newton–Euler model was used to form the dynamic equations and a comparative study is given experiencing major difficulties in using both methods. For development of a robotic system, complete modeling of a well-known robotic manipulator UR5 is presented [10] where the dynamic properties, inertia matrix, coriolis and centrifugal matrix, and gravity vector are derived based on the Lagrange method, and the derived mathematical model has been implemented in MATLAB. In many research article, forward kinematics of a manipulator are discussed [11, 12] with different joint structures such as triangular prism structured links where the varying positions of the end effector were calculated and plotted against joint angles in MATLAB.

In this paper, the dynamic equations of motion have been derived using Newton–Euler formulation. The direct kinematic solution is obtained analytically by expressing the three nonhomogeneous nonlinear second order differential equations into six differential equations of first order and rearranging them in phase variable form. The angular displacement and angular velocities of each link are presented graphically with respect to time for constant torques applied for a specific period of time using MATLAB. Then, the optimum torque ratio is obtained which gives the feasible angular positions of three-links for practical implementations.

2 Three-Link Manipulator

In this work, a robotic manipulator of three-links of lengths l1, l2, l3 and three joints J1, J2, J3 is shown in Fig. 1. The links are having mass m1, m2, m3 and link parameters are α1 = α2 = α3 = 0. All the revolving rigid joints can rotate in a two-dimensional space about the z-axis. The link parameters to be considered as α1 = α2 = α3 = 0, and θ1, θ2, and θ3 are considered as angular displacement of each joint-link pair of the system.

Fig. 1
figure 1

Three-link manipulator with coordinate assignment

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The initial conditions can be considered as

$$ \omega_{0} = 0,\dot{\omega }_{0} = 0,v_{0} = 0 $$

where \(\omega_{0}\) is the initial angular velocity, \(\dot{\omega }_{0}\) is the initial angular acceleration, \(v_{0}\) is the initial linear velocity, and \(\dot{v}_{0}\) is the initial linear acceleration.

Therefore,

$$ \dot{v}_{0} = \left[ {\begin{array}{*{20}c} {g_{x} } \\ {g_{y} } \\ {g_{z} } \\ \end{array} } \right] $$
$$ \dot{v}_{0} = \left[ {\begin{array}{*{20}c} 0 \\ g \\ 0 \\ \end{array} } \right] $$
$$ g = 9.8\;{\text{m}}/{\text{s}}^{2} $$

Joint variables: angular position, angular speed, and angular acceleration

$$ q_{i} = \theta_{i} = \left[ {\theta_{1} ,\theta_{2} ,\theta_{3} } \right]; $$
$$ \dot{q}_{i} = \dot{\theta }_{i} = \left[ {\dot{\theta }_{1} ,\dot{\theta }_{2} ,\dot{\theta }_{3} } \right]; $$
$$ \ddot{q}_{i} = \ddot{\theta }_{i} = \left[ {\ddot{\theta }_{1} ,\ddot{\theta }_{2} ,\ddot{\theta }_{3} } \right]; $$

Link variable:

$$ F_{i} ,f_{i} ,n_{i} ,\tau_{i} $$

where

Fi: Total external force exerted on link i at the center of mass

fi: Force exerted on link i by link i − 1

ni: Moment exerted on link i by link i − 1

τi: Torque applied to joint i.

3 Computed Joint Torque by Newton–Euler Equations of Motion

The Newton–Euler approach has been adopted to compute the kinematic model relating joint torque and angular displacement as given in (1) to (3).

Joint torque computed to each joint actuator for link i = 3, 2, 1 [for link3, link2, link1 are \(\tau_{3} ,\tau_{2} ,\tau_{1}\), respectively]

$$ \begin{aligned} \tau_{3} & = \frac{1}{2}m_{3} l_{1} l_{3} \left[ {\cos \left( {\theta_{2} + \theta_{3} } \right)\ddot{\theta }_{1} + \sin \left( {\theta_{2} + \theta_{3} } \right)\dot{\theta }_{1}^{2} } \right] \\ & \quad + \frac{1}{2}m_{3} l_{2} l_{3} \left[ {\sin \theta_{3} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} } \right)^{2} + \cos \theta_{3} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right)} \right] \\ & \quad + \frac{1}{3}m_{3} l_{3}^{2} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right) + \frac{1}{2}m_{3} l_{3} g\cos \left( {\theta_{1} + \theta_{2} + \theta_{3} } \right) \\ \end{aligned} $$
(1)
$$ \begin{aligned} \tau_{2} & = \frac{1}{2}m_{3} l_{1} l_{3} \left[ {\cos \left( {\theta_{2} + \theta_{3} } \right)\ddot{\theta }_{1} + \sin \left( {\theta_{2} + \theta_{3} } \right)\dot{\theta }_{1}^{2} } \right] \\ & \quad + \frac{1}{2}m_{3} l_{2} l_{3} \left[ {\sin \theta_{3} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} } \right)^{2} + \cos \theta_{3} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right)} \right] \\ & \quad + \frac{1}{3}m_{3} l_{3}^{2} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right) + \frac{1}{2}m_{3} l_{3} g\cos \left( {\theta_{1} + \theta_{2} + \theta_{3} } \right) \\ & \quad + \left( {m_{3} + \frac{1}{2}m_{2} } \right)l_{1} l_{2} \left[ {\cos \theta_{2} \ddot{\theta }_{1} + \sin \theta_{2} \dot{\theta }_{1}^{2} } \right] \\ & \quad + \left( {m_{3} + \frac{1}{3}m_{2} } \right)l_{2}^{2} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right) \\ & \quad + \frac{1}{2}m_{3} l_{2} l_{3} \left[ {\cos \theta_{3} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right) - \sin \theta_{3} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} + \dot{\theta }_{3} } \right)^{2} } \right] \\ & \quad + \left( {m_{3} + \frac{1}{2}m_{2} } \right)l_{2} g\cos \left( {\theta_{1} + \theta_{2} } \right) \\ \end{aligned} $$
(2)
$$ \begin{aligned} \tau_{1} & = \frac{1}{2}m_{3} l_{1} l_{3} \left[ {\cos \left( {\theta_{2} + \theta_{3} } \right)\ddot{\theta }_{1} + \sin \left( {\theta_{2} + \theta_{3} } \right)\dot{\theta }_{1}^{2} } \right] \\ & \quad + \frac{1}{2}m_{3} l_{2} l_{3} \left[ {\sin \theta_{3} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} } \right)^{2} + \cos \theta_{3} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right)} \right] \\ & \quad + \frac{1}{3}m_{3} l_{3}^{2} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right) + \frac{1}{2}m_{3} l_{3} g\cos \left( {\theta_{1} + \theta_{2} + \theta_{3} } \right) \\ & \quad + \left( {m_{3} + \frac{1}{2}m_{2} } \right)l_{1} l_{2} \left[ {\cos \theta_{2} \ddot{\theta }_{1} + \sin \theta_{2} \dot{\theta }_{1}^{2} } \right] \\ & \quad + \left( {m_{3} + \frac{1}{3}m_{2} } \right)l_{2}^{2} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right) \\ & \quad + \frac{1}{2}m_{3} l_{2} l_{3} \left[ {\cos \theta_{3} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right) - \sin \theta_{3} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} + \dot{\theta }_{3} } \right)^{2} } \right] \\ & \quad + \left( {m_{3} + \frac{1}{2}m_{2} } \right)l_{2} g\cos \left( {\theta_{1} + \theta_{2} } \right) + \left( {m_{3} + m_{2} + \frac{1}{3}m_{1} } \right)l_{1}^{2} \ddot{\theta }_{1} \\ & \quad + \left( {m_{3} + \frac{1}{2}m_{2} } \right)l_{1} l_{2} \left[ {\cos \theta_{2} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right) - \sin \theta_{2} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} } \right)^{2} } \right] \\ & \quad - \frac{1}{2}m_{3} l_{1} l_{3} \left[ {\sin \left( {\theta_{2} + \theta_{3} } \right)\left( {\dot{\theta }_{1} + \dot{\theta }_{2} + \dot{\theta }_{3} } \right)^{2} - \cos \left( {\theta_{2} + \theta_{3} } \right)\left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right)} \right] \\ & \quad + \left( {m_{3} + m_{2} + \frac{1}{2}m_{1} } \right)l_{1} g\cos \theta_{1} \\ \end{aligned} $$
(3)

4 Kinematic Solution of Mathematical Modeling of N–E Model of Three-Link Manipulator

As the dynamics relating joint torque and angular displacement of three coupled joint-link are highly nonlinear in nature, a phase variable form has been adopted. In order to derive a systematic procedure, the three nonlinear differential equation of order two is transformed into a phase variable form representing a system of six first order differential equations.

Assuming x1, x2, x3, x4, x5, and x6 are six phase variables given by

$$ x_{1} = \dot{\theta }_{1} $$
(4)
$$ x_{2} = \dot{\theta }_{2} $$
(5)
$$ x_{3} = \dot{\theta }_{3} $$
(6)
$$ x_{4} = \theta_{1} $$
(7)
$$ x_{5} = \theta_{2} $$
(8)
$$ x_{6} = \theta_{3} $$
(9)

Therefore,

$$ \dot{x}_{1} = \ddot{\theta }_{1} $$
(10)
$$ \dot{x}_{2} = \ddot{\theta }_{2} $$
(11)
$$ \dot{x}_{3} = \ddot{\theta }_{3} $$
(12)
$$ \dot{x}_{4} = \dot{\theta }_{1} = x_{1} $$
(13)
$$ \dot{x}_{5} = \dot{\theta }_{2} = x_{2} $$
(14)
$$ \dot{x}_{6} = \dot{\theta }_{3} = x_{3} $$
(15)

Nondimensionalizing (1), (2), and (3), we have

$$ \begin{aligned} \tau_{3} & = \frac{1}{2}\cos \left( {\theta_{2} + \theta_{3} } \right)\ddot{\theta }_{1} + \frac{1}{2}\sin \left( {\theta_{2} + \theta_{3} } \right)\dot{\theta }_{1}^{2} \\ & \quad + \frac{1}{2}\sin \theta_{3} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} } \right)^{2} + \frac{1}{2}\cos \theta_{3} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right) \\ & \quad + \frac{1}{3}\left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right) + \frac{1}{2}\cos \left( {\theta_{1} + \theta_{2} + \theta_{3} } \right) \\ \end{aligned} $$
(16)
$$ \begin{aligned} \tau_{2} & = \frac{1}{2}\cos \left( {\theta_{2} + \theta_{3} } \right)\ddot{\theta }_{1} + \frac{1}{2}\sin \left( {\theta_{2} + \theta_{3} } \right)\dot{\theta }_{1}^{2} \\ & \quad + \frac{1}{2}\sin \theta_{3} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} } \right)^{2} + \frac{1}{2}\cos \theta_{3} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right) \\ & \quad + \frac{1}{3}\left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right) + \frac{1}{2}\cos \left( {\theta_{1} + \theta_{2} + \theta_{3} } \right) \\ & \quad + \frac{3}{2}\cos \theta_{2} \ddot{\theta }_{1} + \frac{3}{2}\sin \theta_{2} \dot{\theta }_{1}^{2} + \frac{4}{3}\left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right) \\ & \quad + \frac{1}{2}\cos \theta_{3} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right) - \frac{1}{2}\sin \theta_{3} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} + \dot{\theta }_{3} } \right)^{2} \\ & \quad + \frac{3}{2}\cos \left( {\theta_{1} + \theta_{2} } \right) \\ \end{aligned} $$
(17)
$$ \begin{aligned} \tau_{1} & = \frac{1}{2}\cos \left( {\theta_{2} + \theta_{3} } \right)\ddot{\theta }_{1} + \frac{1}{2}\sin \left( {\theta_{2} + \theta_{3} } \right)\dot{\theta }_{1}^{2} + \frac{1}{2}\sin \theta_{3} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} } \right)^{2} \\ & \quad + \frac{1}{2}\cos \theta_{3} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right) + \frac{1}{3}\left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right) \\ & \quad + \frac{1}{2}\cos \left( {\theta_{1} + \theta_{2} + \theta_{3} } \right) + \frac{3}{2}\cos \theta_{2} \ddot{\theta }_{1} + \frac{3}{2}\sin \theta_{2} \dot{\theta }_{1}^{2} \\ & \quad + \frac{4}{3}\left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right) + \frac{1}{2}\cos \theta_{3} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right) \\ & \quad - \frac{1}{2}\sin \theta_{3} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} + \dot{\theta }_{3} } \right)^{2} + \frac{3}{2}\cos \left( {\theta_{1} + \theta_{2} } \right) + \frac{7}{3}\ddot{\theta }_{1} \\ & \quad + \frac{3}{2}\cos \theta_{2} \left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} } \right) - \frac{3}{2}\sin \theta_{2} \left( {\dot{\theta }_{1} + \dot{\theta }_{2} } \right)^{2} \\ & \quad - \frac{1}{2}\sin \left( {\theta_{2} + \theta_{3} } \right)\left( {\dot{\theta }_{1} + \dot{\theta }_{2} + \dot{\theta }_{3} } \right)^{2} + \frac{1}{2}\cos \left( {\theta_{2} + \theta_{3} } \right)\left( {\ddot{\theta }_{1} + \ddot{\theta }_{2} + \ddot{\theta }_{3} } \right) + \frac{5}{2}\cos \theta_{1} \\ \end{aligned} $$
(18)

Using (16), (17), and (18), \(\ddot{\theta }_{1} ,\ddot{\theta }_{2} ,\;{\text{and}}\;\ddot{\theta }_{3}\) can be formed as algebraic sum of \(\theta_{1} ,\theta_{2} ,\theta_{3}\) and their first derivatives which give all first order nonlinear equations. Therefore, (10) to (15) are expressed as six equations of first order which can be solved graphically using MATLAB.

4.1 Transient Behavior of Link and Joint Movement

Figures 2, 3, and 4 describe the dynamic behavior of ω and θ of each link corresponding to each joint actuation torque, τ1, τ2, and τ3, for different torque ratio applied for a finite length of time (0.5 s). The angular position versus torque characteristics are shown in Figs. 5, 6, and 7 for different torque ratio.

Fig. 2
figure 2

Transient of angular velocity and angular position for τ1, τ2, τ3:1, 1, 1 (in Nm), respectively

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Fig. 3
figure 3

Transient of angular velocity and angular position for τ1, τ2, τ3:4, 2, 1 (in Nm), respectively

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Fig. 4
figure 4

Transient response of angular velocity and angular position for τ1, τ2, τ3:9, 3, 1 (in Nm), respectively

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Fig. 5
figure 5

Angular position versus applied torque (τ1:τ2:τ3::1:1:1 (in Nm); time: 0.5 s)

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Fig. 6
figure 6

Angular position versus applied torque (τ1:τ2:τ3::4:2:1 (in Nm); time: 0.5 s)

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Fig. 7
figure 7

Angular position versus applied torque (τ1:τ2:τ3::9:3:1 (in Nm); time: 0.5 s)

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4.2 Characteristics of Link and Joint Movement with Applied Torque

See Figs. 5, 6, and 7.

5 Results and Discussion

Table 1 represents the results obtained from Figs. 2 to 4 which describe the angular position of each link after 0.5 s due to application of constant torques.

Table 1 Angular position due to application of constant torques applied for 0.5 s
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From Table 1, it is evident that in nondimentionalize condition of the manipulator, applying joint torques in equal proportion are not able to lift up the link1 and link2 as both θ1 and θ2 are negative. For practical realization, it can be assumed that \(0 \le \theta_{1} \le \frac{\pi }{2},\) \(0 \le \theta_{2} \le \frac{\pi }{2},\) and \(0 \le \theta_{3} \le \frac{\pi }{2}\). And if θ2 is negative, it must follow the condition \(\left| {\theta_{2} } \right| \le \theta_{1}\). Therefore, to satisfy this condition the joint torques must be applied in the proportion of τ1:τ2:τ3::9:3:1 or higher (refer Table 1).

6 Conclusion

The dynamic equations, as stated in (1) to (3), describe the nonlinear nature of the system. Here, the open loop response of the system has been studied and result is as expected. Figures 5–7 show the angular rotation versus torque characteristics and the most linear characteristics and hence optimum torque can be obtained for the torque ratios of τ1:τ2:τ3::9:3:1 as shown in Fig. 7 using MATLAB. Moreover, the system can be assumed to have a combination of three inverted pendulum will also be a chaotic system. As a very consequences, the system can be controlled with sliding mode controller as well as with proportional derivative controller which will give the system more stability.