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Abstract

Pole-climbing robots are increasingly needed to carry out high-risk tasks for human beings. A micro pole-climbing robot is designed in this article. A strategy of climbing pole is proposed, which has high precision in each stride. To enable the robot to sample the angle relative to the ground in real time, micro electro mechanical systems (MEMS) three-axis accelerometers are equipped on micro pole-climbing robot. Accelerometer measurements provide an absolute reference for the pitch-and-roll components of the estimated orientation, which are used as feedback input signal of proportional–integral–derivative algorithm. A supporting structure is installed at the joint of each gripper to assist the robot to clamp a pole. The support structure improves the load capacity of the robot. The maximum load of micro pole-climbing robot is 3.5 times its own weight. The climbing pole strategy includes the following sections: a Denavit–Hartenberg model is established and the inverse kinematic solution is analyzed; the flip locomotion is analyzed; and the parameters of K_P , K_I , and K_D in the proportional–integral–derivative control method are obtained according to the Ziegler–Nichols controller. The performance of pole climbing based on micro pole-climbing robot prototype was tested. By using this strategy of climbing pole, the self-continuous climbing with controllable stride is realized, and the angular velocity fluctuation of the five-bar mechanism driven by steering gear is reduced. The average time of a single step is 27 s, the maximum relative error of step distance is 4.6%, and the average relative error is 2.8%. These results confirm that the structure scheme and the strategy of climbing pole are feasible.

Keywords

Five-bar linkage pole-climbing robot kinematic model PID autonomous climbing

Introduction

Aerial work to climb poles,^1

–5 trees,^6,7 or trusses^8
–10 can result in worker injuries^11,12 due to human kinematic and spatial limitations. Pole-climbing robots are therefore increasingly wanted to carry out high-risk tasks for human beings. There are many studies about pole-climbing robots with significant achievements.^{1

–5,6

–10,13

–24} Autonomous climbing is a necessity for these robots as the operators may not see the next target to be grasped by the robots all the times.

RoMeLa Robot & Mech Lab designed a serpentine robot called HyDRAS,¹ which comprises a serial chain of actuated universal joints for pole climbing. HyDRAS can wrap its body around the structure in a helical shape and rotates its body along its own central body axis to roll up the structure. Haynes et al. made a type of legged robot—RiSE V3.² RiSE V3 is used for dynamical, high-speed climbing of a uniformly convex cylindrical structure. It weighs 5.4 kg and is 70 cm long plus a 28 cm tail appendage. Faizal et al. developed a tree-climbing robot to carry its weight by using two servo motors.⁶ The robot can only climb the trees with a diameter not larger than 10 cm. Another climbing robot designed by the Chinese Academy of Forestry can climb trees of multiple diameters.⁷ Taking a modular approach, a biped-climbing robot (Climbot) was constructed.¹³ Climbot implements autonomous climbing with three basic gaits suitable for different occasions. A pole-climbing robot for transmission line was also built. It adsorbs and locates by an electromagnet module, so it can only climb on metal surfaces.³ Zahra et al. made an electrical pole-climbing robot, which has a good spatial member transition ability, but its size is relatively large.⁴ A robot prototype was developed to construct overhead distribution lines in Quebec hydropower stations.⁵

The pole-climbing robot Pobot V2 was based on the innovative principle of rolling self-locking⁸; it uses no energy to maintain itself at a given altitude. There are also a few prototype robots for truss climbing.^9,10,14 Shady3D built by MIT¹⁰ can move along a three-dimensional (3-D) truss structure. The stability of gripping truss is solved, but the shape of truss section is required. The feasible single-step collision-free climbing motion of BiPCRs in space truss was proposed.¹⁵ The 3-D Climber climbing robot developed by Tavakoli et al.¹⁷ is suitable for climbing pipelines. It has a strong maneuverability but with a large size and a slow movement. The robot is designed to crawl or descend with three wheels—two free wheels and one driving wheel.¹⁸

Some robots can do automatic pole climbing. However, there are few discussions on the accuracy of the robot’s climbing steps. This article describes a micro pole-climbing robot (MPCRobot) and proposed a strategy of climbing pole. By using this strategy of climbing pole, the self-continuous climbing with controllable stride is realized, and the angular velocity fluctuation of the five-bar mechanism driven by steering gear is reduced.

Pole-climbing robot architecture

The 3-D model of MPCRobot is shown in Figure 1. It consists of five bar linkages of a ₀, a ₁, a ₂, a ₃, and a ₄; four joints of $j_{0}$ , j ₁, j ₂, and j ₃; two supports of h ₁ and h ₂; and two grippers of g ₁ and g ₂. The functions of $j_{0}$ and j ₃ are to adjust the angle of the robot fuselage and grasp, and the functions of j ₁ and j ₂ are to adjust the stride distance. The supporting structures of h ₁ and h ₂ are formed between the support frame under the gripper and the pole, making MPCRobot stable as a whole.

Figure 1.

3-D model of the micro pole-climbing robot. 3-D: three-dimensional.

The coordinated actions of a ₁, a ₂, and a ₃ determine the stride of the robot during the overturning process, and a ₀ and a ₄ transmit the distance to the two graspers g ₁ and g ₂ without taking part in the determination of the step size. When a ₀ and a ₄ rotate to each other parallel, the distance between g ₁ and g ₂ is exactly equal to the stride determined by a ₁, a ₂, and a ₃ according to the geometric relationship of the parallelogram.

The advantage of this design is that the structure only needs a ₀ and a ₄ to keep parallel with each other and does not require g ₁ and g ₂ to keep strictly perpendicular to the climbing pole, which simplifies the design of the grippers and also reduces the clamping force of the grippers.

Modelling and analysis

In Figure 1, a ₀ and g ₁ and a ₄ and g ₂ are fixed connections, which are represented by l ₁ and l ₂, respectively. The fixed angle $γ_{1}$ is formed between l ₁ and the pole due to the support h ₁. MPCRobot rotates $j_{0}$ , j ₁, and j ₂ to make j ₃ move to Q in order to realize the stride action according to the preset step size D. Rotation j ₃ makes l ₂ reach a certain posture, and then $γ_{2}$ ( $γ_{2} = γ_{1}$ ) between l ₂ and the pole is formed and then D is parallel to the pole, as shown in Figure 2. In the process, S is regarded as the starting point of the motion as the fixed point, and the control process is to adjust the angle of $j_{0}$ , j ₁, and j ₂ to make j ₃ reach the target Q.

Figure 2.

D-H coordinate system of MPCRobot. D-H: Denavit–Hartenberg; MPCRobot: micro pole-climbing robot.

D-H parameter model

The Denavit–Hartenberg (D-H) coordinate system of MPCRobot is established, as shown in Figure 2. The coordinate system of j ₀ is represented by x ₀–z ₀. The coordinate systems of j ₁, j ₂, and j ₃ are established in turn, such as x ₁–z ₁, x ₂–z ₂, and x ₃–z ₃. The parameters are presented in Table 1. $θ_{i} (i = 1, 2, 3)$ is the angle of linkage, D is the distance between j ₀ and j ₃, $a_{i} (i = 1, 2, 3)$ is the length of each common perpendicular line, and ∂ is the angle between the axis of the joint.

Table 1.

Parameters of D-H coordinate system of MPCRobot.

#	θ	a
0–1	$θ_{1}$	a ₁
1–2	$θ_{2}$	a ₂
2–3	$θ_{3}$	a ₃

D-H: Denavit–Hartenberg; MPCRobot: micro pole-climbing robot.

Kinematics analysis

The general transformation matrix of the two joint coordinate systems can be described as follows

T_{i + 1} = [\begin{matrix} \begin{matrix} cos θ_{i + 1} & - sin θ_{i + 1} cos \partial_{i + 1} \\ sin θ_{i + 1} & cos θ_{i + 1} cos \partial_{i + 1} \end{matrix} & \begin{matrix} sin θ_{i + 1} sin \partial_{i + 1} & a_{i + 1} cos θ_{i + 1} \\ - cos θ_{i + 1} sin \partial_{i + 1} & a_{i + 1} sin θ_{i + 1} \end{matrix} \\ \begin{matrix} 0 & sin \partial_{i + 1} \\ 0 & 0 \end{matrix} & \begin{matrix} cos \partial_{i + 1} & d_{i + 1} \\ 0 & 1 \end{matrix} \end{matrix}]

Set $S_{i} = sin θ_{i}$ , $C_{i} = cos θ_{i}$ , $C_{123} = cos (θ_{1} + θ_{2} + θ_{3})$ , $S_{123} = sin (θ_{1} + θ_{2} + θ_{3})$ , $i = 1, 2, 3$ , then the rotation matrices T ₁, $T_{2}$ , and $T_{3}$ can be obtained

\begin{array}{l} T_{1} = [\begin{matrix} C_{1} & - S_{1} & 0 & a_{1} C_{1} \\ S_{1} & C_{1} & 0 & a_{1} S_{1} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ T_{2} = [\begin{matrix} C_{2} & - S_{2} & 0 & a_{2} C_{2} \\ S_{2} & C_{2} & 0 & a_{2} S_{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ T_{3} = [\begin{matrix} C_{3} & - S_{3} & 0 & a_{3} C_{3} \\ S_{3} & C_{3} & 0 & a_{3} S_{3} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \end{array}

Let the D-H coordinate system of the end of a ₃ be N. When the stride distance D is perpendicular to the ground, it is obvious that $p_{y} = p_{z} = 0$ , $p_{x} = D$ . Transformation matrix from end gripper to pedestal can be obtained by equation (3) ²³

\begin{array}{l} N = T_{1} \times T_{2} \times T_{3} \\ = [\begin{matrix} \begin{matrix} C_{123} & - S_{123} \\ S_{123} & C_{123} \end{matrix} & \begin{matrix} 0 & a_{1} C_{1} + a_{2} C_{12} + a_{3} C_{123} \\ 0 & a_{1} S_{1} + a_{2} S_{12} + a_{3} S_{123} \end{matrix} \\ \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} & \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \end{matrix}] \\ = [\begin{matrix} n_{x} & o_{x} & a_{x} & p_{x} \\ n_{y} & o_{y} & a_{y} & p_{y} \\ n_{z} & o_{z} & a_{z} & p_{z} \\ 0 & 0 & 0 & 1 \end{matrix}] = [\begin{matrix} n_{x} & o_{x} & a_{x} & d \\ n_{y} & o_{y} & a_{y} & 0 \\ n_{z} & o_{z} & a_{z} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \end{array}

Inverse kinematics solutions $θ_{1}$ , $θ_{2}$ , and $θ_{3}$

The inverse kinematics of the end effector is solved by using $T_{1}^{- 1}$ left multiplier equation (3), as shown below²⁴

\begin{array}{l} T_{1}^{- 1} N = T_{1}^{- 1} T_{1} T_{2} T_{3} \\ [\begin{matrix} \begin{matrix} C_{23} & - S_{23} \\ S_{23} & C_{23} \end{matrix} & \begin{matrix} 0 & a_{3} C_{23} + a_{2} C_{2} \\ 0 & a_{3} S_{23} + a_{2} S_{2} \end{matrix} \\ \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} & \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \end{matrix}] = [\begin{matrix} \begin{matrix} n_{x} C_{1} + n_{y} S_{1} & o_{x} C_{1} + o_{y} S_{1} \\ n_{y} C_{1} - n_{x} S_{1} & o_{y} C_{1} - o_{x} S_{1} \end{matrix} & \begin{matrix} a_{x} C_{1} + a_{y} S_{1} & p_{x} C_{1} + p_{y} S_{1} - a_{1} \\ a_{y} C_{1} - a_{x} S_{1} & p_{y} C_{1} - p_{x} S_{1} \end{matrix} \\ \begin{matrix} n_{z} & o_{z} \\ 0 & 0 \end{matrix} & \begin{matrix} a_{z} & p_{z} \\ 0 & 1 \end{matrix} \end{matrix}] \end{array}

Assuming $θ_{1}$ is known and setting $m = p_{x} C_{1} + p_{y} S_{1} - a_{1}$ and $n = p_{y} C_{1} - p_{x} S_{1}$ , then equations (5) and (6) can be obtained by equation (4). Equation (7) can be obtained by equations (3) and (4) square sum.

m = a_{3} C_{23} + a_{2} C_{2}

n = a_{3} S_{23} + a_{2} S_{2}

m^{2} + n^{2} + a_{3}^{2} - 2 a_{3} (m C_{23} + n S_{23}) = a_{2}^{2}

Set k = m C_{23} + n S_{23}, then

k = \frac{m^{2} + n^{2} + a_{3}^{2} - a_{2}^{2}}{2 a_{3}}

Equation (9) can be obtained by using the universal formula of trigonometric function. Then $θ_{23}$ can be obtained as shown in equation (10). $θ_{2}$ can be obtained by substituting equation (10) into equation (6) as shown in equation (11). Finally, $θ_{3}$ can be obtained by equations (10) and (11)

m + k {tan}^{2} \frac{θ_{2} + θ_{3}}{2} - 2 n tan \frac{θ_{2} + θ_{3}}{2} + k - m = 0

θ_{23} = θ_{2} + θ_{3} = 2 arctan \frac{n \pm \sqrt{n^{2} - k^{2} + m^{2}}}{m + k}

θ_{2} = arcsin \frac{n - a_{3} sin (θ_{2} + θ_{3})}{a_{2}}

θ_{3} = 2 arctan \frac{n \pm \sqrt{n^{2} - k^{2} + m^{2}}}{m + k} - θ_{2}

In addition, the inverse trigonometric function has only one root in the definition domain $(- \frac{π}{2}, \frac{π}{2})$ , and there is actually another root in the whole coordinate plane $(- π, π)$ , so another root should be complemented. For the cosine function solution θ, another root $π - θ$ should be complemented. For the arc tangent solution θ, root $π + θ$ should be complemented.

In the inverse kinematic solutions, there are two kinds of solutions which should be abandoned: the wrong mathematical solutions and the infeasible solutions of the system. The wrong mathematical solutions occur because of simultaneous squaring steps on both sides of equation (7). This transformation produces some wrong roots. Such wrong solutions are eliminated by bringing back the original equation. The infeasible solutions are beyond the angle range of the actuator, which can be realized in theory but cannot be realized in actual systems. This kind of solutions is eliminated by defining the rotation angle of the steering gear. The above two kinds of solutions should be removed.

Climbing cycle for locomotion

MPCRobot climbing a pole has two gaits: the flipping-over gait and the inchworm gait. The energy consumed with the inchworm gait is the larger than the flipping-over gait.¹³ MPCRobot is used for power line monitoring (with cameras and sensors) and lightning rod detection. This kind of working occasions have enough working space, so this article only discusses the flipping-over gait.

In the process of continuous climbing, the robot’s posture is alternately forward and backward. In this article, the upright posture is defined as the state of g ₁ below g ₂, otherwise defined as an inverted posture. The self-continuous climbing process with controllable strides is decomposed into two stages: flip and stride.

Flip locomotion

The flip locomotion consists of three steps: loosening gripper g ₁, curling up, and turning over, as shown in Figure 3.

Figure 3.

MPCRobot flip locomotion. (a) Loosening gripper g ₁. (b) Curling up. (c) Turning over. MPCRobot: micro pole-climbing robot.

Firstly, gripper g ₁ is loosened, and gripper g ₂ is used as the fixed end of the robot, as shown in Figure 3(a). Secondly, the robot’s joints $j_{0}$ , j ₁, and j ₂ rotate to achieve a curling posture, as shown in Figure 3(b). After the robot curls up, the overall length of the robot is greatly reduced, so as to avoid the instability of the robot caused by the excessive torque of the grasp during the overturning process. Thirdly, the robot turns over by rotating j ₃ in a curled state, as shown in Figure 3(c).

Stride locomotion

The robot continues to perform stride locomotion after the flip locomotion. In the process of continuous climb, the robot will be alternatively in the upright posture and inverted posture. Whether in the upright or inverted postures, the inverse kinematic solution of stride is invariable. The obtained $θ_{1}$ , $θ_{2}$ , and $θ_{3}$ are used to control the rotations of $j_{0}$ , j ₁, and j ₂ (in the upright posture) or the rotations of j ₃, j ₂, and j ₁ (in the inverted posture), respectively.

After experiencing a row of gesture in Figure 3, the robot is in the inverted posture. $θ_{1}$ , $θ_{2}$ , and $θ_{3}$ are used to control the rotations of j ₃, j ₂, and j ₁. The stride locomotion consists of three steps: the determination of the stride parameter D, the adjustment of the stride posture, and the clamping of the gripper g ₁, as shown in Figure 4.

Figure 4.

MPCRobot stride locomotion. MPCRobot: micro pole-climbing robot. (a) generating D. (b) paralleling. (c) grabbing with.

First, inputing the expected stride D and using equations (5) to (11), $θ_{2}$ and $θ_{3}$ can be calculated according to angle $θ_{1}$ . The robot’s joints j ₁ and j ₂ rotate according to $θ_{2}$ and $θ_{3}$ , and then a ₁, a ₂, and a ₃ generate D, as shown in Figure 4(a).

Second, by adjusting j ₃, the robot keeps the line D between j ₀ and j ₃ parallel to the climbing pole, as shown in Figure 4(b). When the climbing pole is perpendicular to the ground and the angle between $a_{3}$ and the ground reaches $θ_{1} + 90$ °, D is exactly parallel to the climbing pole. The real-time angle between a ₃ and the ground is measured by an accelerometer mounted on a ₃.

Finally, the robot adjusts the attitude of the gripper g ₁ by rotating j ₀ and grabbing the climbing pole with g ₁, as shown in Figure 4(c). For climbing robots, it is very important for the gripper to grasp the climbing pole accurately and firmly. Unreliable clamping will cause robot’s falling and serious damages. Traditional calibration schemes use infrared, laser, ultrasound, and other ranging sensors or cameras.^20
–22 By judging the relative position between the gripper and the pole, the robot is guided to realize alignment between gripper and climbing pole. Such schemes require more complex processing and fusion of data from sensors, which increase the cost, size, and weight of robots. This article presents an autonomous gripping scheme based on robot geometry structure, which does not require the use of additional distance measuring sensors. Two supports h ₁ and h ₂ are added to two grippers g ₁ and g ₂ to assist the grippers to grasp the pole, as shown in Figure 6(c). With the help of accelerometers mounted on g ₁ and g ₂, the robot can clamp the vertical climbing pole automatically when two grippers g ₁ and g ₂ are parallel to each other.

Prototype and experiment

Robot prototype

We designed MPCRobot prototype, as shown in Figure 5. The prototype weighs 1.07 kg (including a battery), and it is 600 mm long when fully extended. The specific size is shown in Table 2. Constrained by the robot’s current structure, a single span of the robot is 8–23 cm. According to the results of the force analysis, the TBS2701 steering gear (its main parameters are listed in Table 3) was selected as the joint motor. This motor has a maximum rotation angle of 270° and can operate in joint mode, in which $j_{0}$ , j ₁, j ₂, and j ₃ can be conveniently located at any angle position. MPCRobot realizes an automatic pole climbing.

Figure 5.

MPCRobot prototype. MPCRobot: micro pole-climbing robot.

Figure 6.

Installation diagram of acceleration sensors.

Table 2.

Dimensions of MPCRobot components.

Component	a ₀	a ₁	a ₂	a ₃	a ₄	h ₁	h ₂	g ₁	g ₂
Length (mm)	82	84	81	84	82	40	40	110	110

MPCRobot: micro pole-climbing robot.

Table 3.

Main parameters of TBS2701 steering engine.

Product model	Nominal voltage (V)	Max torque (kg cm)	Nominal speed (s/60°)	Rotate angle	Resolution ratio	Mass (g)	Control mode
TBS2701	5–8.4	15	0.16	0–270°	0.24°	60	PWM

PWM: pulse width modulation.

To enable the robot to sample the angle relative to the ground in real time, micro electro mechanical systems (MEMS) three-axis accelerometers (MPU6050; InvenSense Inc., Sunnyvale, USA) are equipped on $l_{1}, a_{1}$ , a ₂, a ₃, and l ₂, respectively, named $A_{1}, A_{2}, A_{3}, A_{4}, and A_{5}$ , as shown in Figure 6. The accelerometers can measure acceleration along their own axes in the coordinate system. Accelerometer measurements provide an absolute reference for the pitch-and-roll components of the estimated orientation.²⁵ $(x_{A i}, y_{A i}, z_{A i}),$ (i = 1,2,3,4,5) denotes the gravitational acceleration component of $A_{i} (i = 1, 2, 3, 4, 5)$ along its own axes x, y, and z. The full measuring range of the acceleration $A_{i} (i = 1, 2, 3, 4, 5)$ is ±2 g. The measurements $(x_{A i}, y_{A i}, z_{A i}),$ (i = 1,2,3,4,5) derived from $A_{i} (i = 1, 2, 3, 4, 5)$ are used to calculate the current attitude of the robot. $(x_{A 1}, y_{A 1}, z_{A 1})$ is used to adjust $(x_{A 5}, y_{A 5}, z_{A 5})$ by rotating j ₃ to ensure that l ₂ is parallel to l ₁. $(x_{A i}, y_{A i}, z_{A i}) (i = 2, 3, 4)$ is used to adjust the attitude of $a_{i - 1}$ by rotating $j_{i - 2}$ (see section “Joint PID control parameters”).

Joint PID control parameters

Overall system with proposed control strategy

Because the steering gear has a dead zone and a hysteresis in control, some control strategies based on proportional–integral–derivative (PID) control theory are developed to reduce the angular velocity fluctuation of a five-bar mechanism driven by the steering gear.²⁶ Figure 7 shows the overall system with the proposed control strategy, in which the control parameters are the proportional gain K_P _, the integral gain K_I , and the derivative gain K_D . $β_{i}^{′}$ can be obtained by accumulating $θ_{i}$ (i = 1,2,3), which is calculated by the D-H method. $β_{i}^{′}$ can be expressed as follows

β_{k}^{′} = 90 + \sum_{i = 1}^{k} θ_{i} k = (1, 2, 3)

- 3 π + 90 \leq β_{k}^{′} \leq 3 π + 90

Figure 7.

Overall system with the proposed control strategy.

$β_{k}$ is calculated from $(x_{A i}, y_{A i}, z_{A i}) (i = 2, 3, 4)$ by the accelerometers $A_{2}, A_{3}, and A_{4}$ using the angle transformation, which can be presented as

β_{k} = F ({tan}^{- 1} | \frac{x_{A i}}{z_{A i}} |) = {\begin{cases} {tan}^{- 1} | \frac{x_{A i}}{z_{A i}} | x_{A i} > 0, z_{A i} > 0 \\ π - {tan}^{- 1} | \frac{x_{A i}}{z_{A i}} | x_{A i} > 0, z_{A i} < 0 \\ - π + {tan}^{- 1} | \frac{x_{A i}}{z_{A i}} | x_{A i} < 0, z_{A i} < 0 \\ - {tan}^{- 1} | \frac{x_{A i}}{z_{A i}} | x_{A i} < 0, z_{A i} > 0 \\ 0 x_{A i} = 0, z_{A i} > 0 \\ π x_{A i} = 0, z_{A i} < 0 \\ π / 2 x_{A i} > 0, z_{A i} = 0 \end{cases}}

$β_{k}$ is subtracted from $β_{i}^{′}$ , and the resulting error $Δ β$ is sent to the Ziegler–Nichols controller. The parameters of K_P _, K_I , and K_D can be obtained according to the Ziegler–Nichols controller.^27,28 $Δ β$ is presented as

Δ β = β_{k}^{′} - β_{k} + 2 π j j \in {- 2, - 1, 0, 1, 2}

- π \leq Δ β \leq π

PID controller parameters

The stabilization time of each adjustment of the steering gear is 1.2 s. The sampling time of the controller is set to $1.2 s$ . In order for $θ_{1}$ to produce equal amplitude oscillation, let the value of K_c be 4.3. By doing so, an approximate equal amplitude oscillation shown in Figure 8 can be obtained. The oscillation period T_c tends to be stable after T_c ₃, and T_c = 11.5 s can be extracted from Figure 8. According to Table 4,²⁹ the parameters of PID are K_P = 2.6, K_I = 0.45, and K_D = 3.59.

Figure 8.

Approximate equal amplitude oscillation under K_c = 4.3.

Table 4.

Main parameters of the steering engine.

Controller	Calculating characteristic parameters
Controller	K_P	T_n	T_v	K_I	K_D
P	0.5 K_c	—	—	—	—
PD	0.8 K_c	—	0.12 T_c	—	K_P•T_v
PI	0.45 K_c	0.85 T_c	—	$\frac{K_{p}}{T_{n}}$	—
PID	0.6 K_c	0.5 T_c	0.12 T_c	$\frac{K_{p}}{T_{n}}$	K_P•T_v

PID: proportional–integral–derivative.

PID testing

Experimental conditions are as follows: the angle $θ_{1}$ is set to 128°, and the steady-state errors of $θ_{1}$ with and without PID are shown in Figure 9. Mean absolute deviation with and without PID are 0.262 and 0.350, respectively. It shows that the precision of PID control is higher than proportional control. Standard deviation with and without PID are 0.348 and 0.446, respectively. It can be seen that the stability of the PID algorithm is improved. The proportion of steady-state error of using PID within (±0.4°) is increased, which is 58% higher than that without PID algorithm. The average execution time is about 17 s.

Figure 9.

Comparison of steady-state error with and without PID algorithm. PID: proportional–integral–derivative.

When $θ_{1}$ reaches the set angle, it produces the jitter of the robot due to the continuous fine-tune in the range of a steady-state error. Therefore, the rule is set as follows: when the measured steady-state error is less than 0.3° for two consecutive actions, the adjustment is ended.

Clamping force of the grippers

Gripper force analysis mode

The grippers need to fix the robot effectively on the pole to avoid sliding. The force analysis model of gripper is derived based on diagram, as shown in Figure 10. The gripper has four main force points of p ₁, p ₂, p ₃, and p ₄. p ₁ and p ₂ are subjected to upward friction forces of F_S ₁ and F_S ₂ and outward support forces of F_N ₁ and F_N ₂ perpendicular to the rod, respectively. p ₃ and p ₄ are subjected to upward friction forces of F_S ₃ and F_S ₄ and outward support forces of F_N ₃ and F_N ₄ perpendicular to the rod, respectively. F₃₁ and F ₄₂ are the friction forces when the gripper rotates clockwise with respect to the pole. F ₃₂ and F ₄₁ are generated when the gripper rotates counterclockwise. F ₃₁ and F ₄₂ prevent the gripper from turning clockwise. F ₃₂ and F ₄₁ prevent the gripper from turning counterclockwise.

Figure 10.

Gripper force analysis.

In order to stay on pole, equation (18) can be obtained

G < F_{S 1} + F_{S 2} + F_{S 3} + F_{S 4}

F_{S 1} = μ_{S} F_{N 1}

F_{S 2} = μ_{S} F_{N 2}

F_{S 3} = μ_{S} F_{N 3}

F_{S 4} = μ_{S} F_{N 4}

where G is the gravity of the robot at the center of mass m and μ_S is the static coefficient of friction between gripper and pole. The friction forces of F_S ₁, F_S ₂, F_S ₃, and F_S ₄ increase with the increases of F_N ₁, F_N ₂, F_N ₃, and F_N ₄ which are related to the clamping force of the gripper. Above is the static analysis of the gripper force, while the dynamic analysis process is more complex.

Physical testing

In physical testing, the clamping forces of the gripper to hold the pole with different materials (wood, iron, polyvinyl chloride (PVC), and PVC with rubber pad) are tested. The gripper is made of aluminum alloy material. The maximum tension angle of gripper is 55 cm. The maximum torque of the motor driving the gripper is 15 kg cm. The experimental process is as follows: the lower gripper of the robot is clamped, the upper gripper is suspended, and the vertical downward pulling force F is applied at p ₂. F was measured with a tensiometer (Edinburgh Instrument, HP-500, indexing value 0.1 N). Increasing F until the lower gripper slips from the climbing rod, and record the value of the pulling force (called critical value of F) at this time. The experimental data are shown in Table 5.

Table 5.

Results of anti-slip load capacity test.

Material	Diameter (mm)	Critical value of F (N)
PVC	24	21.1
Iron	24	49.2
PVC (with rubber pad)	24	52.3
Wood	24	96.6

In the test, the clamping experiments of pole climbing with the same diameter, but different materials were carried out. The critical force of the gripper on PVC material is 21.1 N. In order to increase the critical force on the PVC pole, a layer of rubber was wrapped around the PVC pole. From Table 5, the addition of rubber layer did increase the critical value from 21.2 N to 52.3 N, which is due to the increase of sliding friction coefficient between the materials. The experimental results show that increasing the friction coefficient between the gripper and the rod can improve F, which means an increased load capacity of the overturning robot.

Pole-climbing experiment

This article designs an experiment to test the climbing speed and the accuracy of step distance, respectively. On a vertical climbing pole, the robot makes repeated strides with a stride of 8–23 cm. The self-continuous climbing process with controllable stride is shown in Figure 11. The flip locomotion, that is, loosening gripper g ₁, curling up, and turning over, is shown in Figure 11(a), (b) and (c), respectively. The stride locomotion, that is, the determination of the stride parameter D, the adjustment of the stride posture, and the clamping of the gripper g ₁, is shown in Figure 11(e), (f) and (g), respectively.

Figure 11.

Continuous climbing steps of MPCRobot. MPCRobot: micro pole-climbing robot. (a) Loosening gripper. (b) Curling up. (c) Turning over. (d) generating D. (e) paralleling. (f) grabbing with.

The time consumed and the errors of stride are recorded, respectively.

Stride experiment

The measurement started with a curl-up (Figure 11(c)) and ended with grabbing the climbing pole with g ₁ (Figure 11(f)). In the physical testing, a single stride was 8–23 mm, and the step length was 1 mm. Sixteen groups of experiments were conducted, and each group was repeated 10 times. Statistics of time consumed for single stride are given in Figure 12. “Percentage” in Figure 12 represents the percentage of the total number of experiments that took time in the abscissa range, and “cumulative percentage” represents the percentage of all results that were less than or equal to the total number of experiments that took time in the abscissa range.

Figure 12.

Single stride time chart.

In the test, the step length of each group of experiments was different (from 8 mm to 23 mm), which resulted in different execution time for the robot to complete the step action. The proportion of complete action within 40 s was close to 90%, and the average time consumed was about 27 s.

Flip experiment

When each stride is completed, the distance between two grippers is measured with a ruler. Compared with the set distance, the relative error statistics of the single stride can be obtained, as shown in Figure 13. “Percentage” in Figure 1 indicates the percentage of errors in the abscissa range to the total number of experiments, and “cumulative percentage” represents the percentage of all results less than or equal to the total number of current abscissa errors.

Figure 13.

Statistics of relative error of single stride.

The range of a single stride is 8–23 cm. The maximum relative error is 4.6%, and the average error is 2.8%. The single span of MPCRobot is 8–23 cm. The absolute error of a single span is 0.22–0.64 cm.

Comparison of climbing robot

A comparison of climbing robot is shown in Table 6. Pole size indicates the cross section of the climbing pole. The pole size that MPCRobot can clamp is the smallest so as to tighten the pole with smaller diameter. The maximum load of Treebot is 2.9 times its own weight, while the maximum load of MPCRobot is 3.5 times its own weight. This is due to the triangular structure consisting of a gripper, a support structure, and a climbing pole (see Figure 1), which can provide stable support for the robot. The single span of MPCRobot is 8–23 cm, and relative error of stride is 2.8%. Then the absolute error of a single span is 0.22–0.64 cm. MPCRobot striding has high accuracy in each step length. Thus, MPCRobot can be used to climb small diameter poles to complete fixed-point monitoring tasks, such as power line monitoring (with cameras and sensors), truss climbing and lightning rod detection, and other working occasions.

Table 6.

Comparison of main parameters of robot.

	Mass (kg)	Full expansion length (cm)	Max load (kg)	Max speed (cm/min)	Relative error of stride (%)	Pole size (mm)
Pipelinebot³⁰	0.46	22	—	67.2	—	Φ50–60
Treebot²¹	0.6	66.5	1.75	22.4	—	Φ64–452
Minibot³¹	0.75	49	—	—	—	Φ40
MPCRobot	1.07	60	3.79	48	2.8	Φ8–55
Shady3D¹⁰	1.34	25	—	20.4	—	19 × 19 (Rectangular cross section)
Climbot¹³	17.5	152	—	220	—	Φ50–120
3DClimber¹⁷	42	60	—	100	—	Φ200–350

Conclusions

A 3-D model of MPCRobot was presented. It consists of five-bar linkages, four joints, two supports, and two grippers. A supporting structure was installed at the joint of each gripper to assist the robot to clamp the climbing pole. The support structure improves the load capacity of the robot. The maximum load of MPcrobot is 3.5 times its own weight. To enable the robot to sample the angle relative to the ground in real time, MEMS three-axis accelerometers are equipped on MPCRobot. Accelerometer measurements provide an absolute reference for the pitch and roll components of the estimated orientation, which are used as feedback input signal of PID algorithm.

A strategy of climbing pole is proposed, which has high precision in each stride. The climbing pole strategy includes the following sections: a D-H model is established and the inverse kinematic solution is analyzed; the flip locomotion is analyzed; and the parameters of K_P _, K_I , and K_D in the PID control method are obtained according to the Ziegler–Nichols controller.

The MPCRobot prototype was thus developed and tested for pole climbing. By using this strategy of climbing pole, the self-continuous climbing with controllable stride is realized, and the angular velocity fluctuation of the five-bar mechanism driven by steering gear is reduced. The average time of a single step is 27 s, the maximum relative error of step distance is 4.6%, and the average relative error is 2.8%. These results confirm that the structure scheme and the strategy of climbing pole are feasible.

Footnotes

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.

ORCID iD

Du Qiaoling

References

Goldman

Hong

. Considerations for finding the optimal design parameters for a novel pole climbing robot. In: 32nd Annual mechanisms and robotics conference, New York, USA, 03–06 August 2008.

Haynes

Khripin

Lynch

. Rapid pole climbing with a quadrupedal robot. In: IEEE International conference on robotics and automation, Kobe, Japan, 12–17 May 2009.

Song

Wan

. Research of underactuated gripper for power tower climbing robot. Chinese J Sci Instru 2015; 36(Supplement): 6.

Zahra

Hasib

Razu

. Electrical Pole climbing robot for wiring and repairing distribution lines. In: 18th International conference on computer and information technology, Dhaka, Bangladesh, 21–23 December 2015.

Allan

Lavoie

Reiher

. Climbing and pole line hardware installation robot for construction of distribution lines. In: 1st International conference on applied robotics for the power industry, Montreal, CANADA, 05–07 October 2010.

Faizal

MIN

Othman

WAFW

Hassan

SSNA

. Development of pole-like tree climbing robot. In:5th IEEE international conference on control system, computing and engineering (ICCSCE), Batu Ferringhi, Malaysia, 27–29 November 2015.

Cheng

. Design and Implementation of a new type of climbing robot for Living trees. Wood Proc Mach 2016; 27(01): 32–34+56.

Fauroux

Morillon

. Design of a climbing robot for cylindro-conicpoles based on rolling self-locking. Ind Robot 2010; 37(3): 287–292.

Zhu

Guan

. Evaluation of graspable region and selection of footholds for biped pole-climbing robots. In: 2014 IEEE international conference on robotics and biomimetics (ROBIO), Bali, Indonesia, 05–10 December 2014.

10.

Yoon

Rus

. Shady3D: a robot that climbs 3D trusses. In: IEEE international conference on robotics and automation, Rome, Italy, 10–14 April 2007.

11.

Zhang

Liu

. Review on key technologies of rod climbing robot. Comput Meas Control 2018; 26(01): 1–5.

12.

Chen

Cao

Yang

. Research status and outlook of rod climbing robots. Ordnanc Indus Autom 2018; 37(03): 21–31.

13.

Guan

Jiang

Zhu

. Climbot: a bio-inspired modular biped climbing robot—system development, climbing gaits, and experiments. J Mech Robot 2016; 8(2).

14.

Chen

Zhu

. Representation of truss-style structures for autonomous climbing of biped pole-climbing robots. Robot Auton Syst 2018; 101: 126–137.

15.

Chen

Zhu

Guan

. Collision-free single-step motion planning of biped pole-climbing robots in spatial trusses. In: IEEE International conference on robotics and biomimetics (ROBIO), Shenzhen, China, 12–14 December 2013.

16.

Zhou

Jiang

Guan

. Energy-optimal motion planning of a biped pole-climbing robot with kinodynamic constraints. Industr Robot Int J Robot Res Appl 2018; 45(3): 343–353.

17.

Tavakoli

Marques

Almeida

ATD

. 3DCLIMBER: climbing and manipulation over 3D structures. Mechatronics 2011; 21(1): 48–62.

18.

Sadeghi

Moradi

Ahmadabadi

. Analysis, simulation, and implementation of a human-inspired pole climbing robot. Robotica 2012; 30: 279–287.

19.

Tavakoli

Zakerzadeh

Vossoughi

. A hybrid pole climbing and manipulating robot with minimum DOFs for construction and service applications. Ind Robot 2005; 32(2): 171–178.

20.

Tavakoli

Marques

De Almeida

. Self calibration of step-by-step based climbing robots. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, St Louis, MO, USA, 10–15 October, 2009, pp. 3297–3303.

21.

Lam

. Treebot: autonomous tree climbing by tactile sensing. Int Conf Robot Autom 2011; 789–794.

22.

Tavakoli

Marques

De Almeida

. A low-cost approach for self-calibration of climbing robots. Robotica 2011; 29(01): 23–34.

23.

Guo

Cui

. Inverse kinematics of 6-DOF manipulator. Agricult Mech Res 2016; 38(12): 51–55.

24.

Guo

Cao

. Innovative closed-form solution and workspace analysis for a 5-DOF manipulator. J Huazhong U Sci (Natural Science Edition) 2015; 43(S1): 14–18.

25.

Chen

Zhao

. Motion recognition based on a three-axis accelerometer. In: Proceedings of the 32nd Chinese control conference, Xi’an, China. 26–28 July 2013.

26.

Wan

Guo

. A steering gear system controller in missiles based on the fractal modified PID algorithm. J Xidian Univ 2007; 34(1): 145–148.

27.

Reis

MRC

Silva

Araujo

WRH

. Speed control for direct current moto r using optimization tuning for PID controller. In: 2016 - International conference on environment and electrical engineering, Florence, Italy, 7–10 June 2016.

28.

Oveisi

Gudarzi

. Adaptive sliding mode vibration control of a nonlinear smart beam: a comparison with self-tuning Ziegler-Nichols PID controller. J Low Freq Noise V A 2013; 32(1): 41–62.

29.

Dong

. PID controller parameter tuning based on Ziegler-Nichols rule. Autom Instrument 2015; (07): 105–106.

30.

Luo

. Analysis of dynamic characteristics and structural design of pipeline climbing robot. Wuhan: Wuhan University of Science and Technology, 2015.

31.

Cai

Zhu

Jiang

. A biologically inspired miniature biped climbing robot. In: International conference on mechatronics and automation, Changchun, China, 9–12 August 2009, pp. 2653–2658. IEEE.

Design of a micro pole-climbing robot

Abstract

Keywords

Introduction

Pole-climbing robot architecture

Modelling and analysis

D-H parameter model

Kinematics analysis

Inverse kinematics solutions θ 1 , θ 2 , and θ 3

Climbing cycle for locomotion

Flip locomotion

Stride locomotion

Prototype and experiment

Robot prototype

Joint PID control parameters

Overall system with proposed control strategy

PID controller parameters

PID testing

Clamping force of the grippers

Gripper force analysis mode

Physical testing

Pole-climbing experiment

Stride experiment

Flip experiment

Comparison of climbing robot

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Inverse kinematics solutions $θ_{1}$ , $θ_{2}$ , and $θ_{3}$