Load capacity of a new rope-climbing robot

. Theoretical analysis and experimental research are carried out on the load capacity of one new kind rope-climbing robot, this robot could lift load along the rope depends on the friction between rope and wheel of the robot. The classic Euler formula is based on the assumption of constant friction coefficient, and this paper establishes a new load capacity model based on non-constant friction coefficient model, presents a method to measure the distribution of friction coefficient, and builds the experimental platform to conduct experimental research on the friction coefficient and load capacity. Experimental results show that the friction coefficient decreases with the increase of wrap angle; the new model fits the results better than Euler formula.


Introduction
Friction transmission is widely used in civil and industrial fields, mine hoist, elevator and belt drive machine are some common typical forms of friction transmission [1][2][3] , the rope-clibing robot to be studied in this paper is another application of friction transmission. The ropeclimbing robot transports loads along the rope relied on the friction between its wheel and rope, and load capacity is a very important measure of the performance of the robot. For this kind rope-climbing robot, its load capacity equals to the maximum friction between rope and wheel, Euler formula is widely used to calculate the friction between wheel and rope or blet or other forms, but there is some deviation between the actual force and calculated values, especially for the rope-climbing robot. Most studies had applied Euler formula to different areas [1][2][3][4][5] , some studies had modified Euler formulation for their special materials and structures [6,7] , but it's assumed that the friction coefficient between rope and wheel keeps constant in these studies, the assumption is too simplified to correspond to reality. The rope-climbing robot as the research object, the load capacity of the robot and the distribution rules of friction coefficient between rope and wheel will be studied in this paper.
In the equation, f  , M  , N  are the friction, centrifugal force and support force acting on the microsegment rope,   is the friction coefficient between the rope and wheel surface at this position and moment,  is the quality of the rope per unit length. Suppose that the friction coefficient is constant over the entire wrap angle [8] ,which could be described by formula =    , Eq. (3) is called Euler's formula because it was deduced by Euler firstly, as the classical formula for calculating the belt (or rope) -wheel friction, it is widely used in mine hoist, elevators, automotive belt transmission and so on. However, the mechanical properties of the rope used for the climbing robot is special, Coulomb friction hypothesis does not hold in the global context, so it is necessary to analyze the problem under the premise of non-constant friction coefficient .

Theoretical analysis based on non-constant friction coefficient
Although the friction coefficient is not constant in the global context, but focus on the micro-segment rope at the very moment, the local friciton coefficient could be seen as a constant and the relationship between the forces at both ends of the micro-segment could be described accurately by Euler Formula. Divide evenly the entire wrap angle to n copies, mark the friction coefficient in number i wrap angle From Eq.(4), the friction coefficients satisfy the following equations: As Eq.(6) shows, the average friction coefficient of a wrap angle from beginning numerically equals to the equal weighted average of friction coefficients of all micro-segments rope, and it is applicable for the average friction coefficient of any wrap angle anywhere. Expand the equation j F : , , Eq.(11) could be used to calculate the load capacity under the condition of non-constant friction coefficient, and Eq.(14) is the theoretical inference of the friction coefficient distribution.

Experimental Research
The theoretical derivation is given above; measure the static friction of different wrap angles in experiments, the function of friction coefficient along wrap angle could be obtained by computing, fitting and deriving the data. In actual use, the end tension (initial force) would be larger than 1 N, and its curve is part of the complete curve. Fig.2 shows the experiment platform, in the experiments, end tension F B would be applied by weights hung at one end of the rope, the signal of A F is received by mechanical sensor fixed at the other end, and then transferred by a transmitter to PC. The experimental procedure is as follows: 1. Start the motor a few seconds after the weights hung stably; 2. Increase the motor's speed gradually until the rope slips globally in the wheel surface; 3. Shut down the motor. Record the corresponding F A signal of whole process above. Use a new rope segment for each test to avoid the effects of wear on the test results, repeat the test at least 15 times under the same conditions, the average F A obtained from 15 values is seen as the load capacity under this condition. Change the wrap angle by adjusting the position of the sensor along the hanging ring rail on the fixture, and measure load capacity under new conditions. Draw the curve of load capacity changed with wrap angle, the results are shown in Fig.4. It is obvious that the load capacity and the wrap angle are approximately exponentially related, but the error will become larger as the wrap angle increases; the average coefficient of static friction decreases with the wrap angle increasing. It can be inferred that the function ( )   decreases monotonically with the wrap angle and tends to a constant value, that is, its first derivative should be negative and gradually approach 0; and its integral function should increase monotonically according to its physical meaning, construct a ( )   function of the following form: (15) In the above formula, there is a>0, b<0, c>0. Select the appropriate parameter, the error between model predictive value and actual value is very small, it indicates that when the wrap angle increases to a certain extent, the effect of wrap angle on friction is very slight,

New model of load capacity
According to Eq.(13) and Eq.(14), the curve in Fig.5 is just a part of the complete curve starts from point (0,1). On the complete curve, the load force corresponds to the wrap angle, as shown in Fig.6, a force F B corresponds to angle B  , and the warp angle is  , the start point of the integral is ( , )

Conclusion
Theoretical derivation and experimental analysis are carried out on the load capacity of a new kind of ropeclimbing robot, mathematic model of the load capacity is obtained considering non-constant friction coefficient, and a measuring method of the friction coefficient distribution is proposed. The experimental results show the friction coefficient decreases with the increase of wrap angle; and the new model has higher precision than Euler formula.