CN103753534B - A kind of Movement Controller of Mobile Robot and control method thereof - Google Patents

Abstract

The invention discloses a kind of Movement Controller of Mobile Robot and control method, comprise remote controller, radio receiving transmitting module, industrial computer, receiver, sensor, wireless transmission system of sensor, proportioning valve, switch valve; Gather the control information of telecontrol panel by remote controller collection plate, send to main industrial computer with instruction type, industrial computer gathers the information of each joint absolute value encoder simultaneously, after main industrial computer carries out Kinematic Algorithms processing, control instruction is sent to receiver, complete the motion control of mobile assembly robot. Utilization of the present invention is connected in its end specific purpose tool, adopting principal and subordinate to add autonomous control mode replaces people's arm to complete the high automobile assembling task of frequency, can alleviate operating personnel's labour intensity on the one hand, on the other hand also for operating personnel's safety provides the mobile assembly robot controller and the control method that ensure.

Description

A kind of Movement Controller of Mobile Robot and control method thereof

Technical field

The present invention relates to a kind of Robotics, especially a kind of Movement Controller of Mobile Robot and control method thereof.

Background technology

Assembly robot is the nucleus equipment of flexible automation assembly system, by robot manipulator, controller, end effectorForm with sensor-based system. Wherein the structure type of manipulator has flapping articulation type, cartesian co-ordinate type, joint type and circular cylindrical coordinateType etc.; General many CPU or the multi-level computer system of adopting of controller, realizes motion control and movement programming; End effector isAdapt to different assembling objects and be designed to various paws and wrist etc.; Sensor-based system obtains again put together machines human and environment and dressInteractional information between pairing resembles. It is PUMA machine that conventional assembly robot mainly contains general programmable assembly manipulation handPeople (come across the earliest 1978, the ancestral of industrial robot is begun) and plane doublejointed humanoid robot are SCARA robot two kindsType. Compared with general industry robot, assembly robot has that precision is high, compliance good, working range is little, can be with otherSystem such as supports the use at the feature, is mainly used in the manufacturing of various electrical equipment.

Current assembly robot is all solid mechanical arm, mechanical arm is arranged on mobile robot, therefore with fixing machineryArm is compared, and has the flexibility of larger working space and Geng Gao, can obtain applying more widely. But mobile assembly machineDevice people is more complicated than the structure of fixing operation arm, so also more difficult to operations such as the planning of its end, controls.

Summary of the invention

The object of the invention is for overcoming above-mentioned the deficiencies in the prior art, a kind of mobile assembly robot controller is provided, utilize connectionBe connected on its end specific purpose tool, the arm that adopts principal and subordinate to add autonomous control mode replacement people completes the assembling work of automobile, on the one handCan alleviate operating personnel's labour intensity, on the other hand also for operating personnel's safety provides the mobile assembly robot control ensureingDevice processed and control method.

For achieving the above object, the present invention adopts following technical proposals.

A kind of Movement Controller of Mobile Robot, comprises remote controller, radio receiving transmitting module, industrial computer, receiver, sensor, sensingDevice wireless transmitting system, proportioning valve, switch valve.

Described remote controller is communicated by letter with industrial computer by wireless module, and industrial computer connects receiver, receiver and proportioning valve, switch valveConnect, sensor is located in robot, and connects industrial computer by wireless transmission system of sensor, the instruction of described remote controller,Be handed down to industrial computer by wireless module, industrial computer passes through the instruction of receiver downloading speed, switch drive instruction to robot; InstituteState remote controller comprise analogue collection module, digital data acquisition module, magnetic lotus root isolation module I, magnetic lotus root isolation module II, magnetic lotus root everyFrom module ii I, reset circuit, jtag circuit, microcontroller circuit, serial port drive circuit. Analogue collection module, digital quantityAcquisition module is connected with microprocessor by magnetic lotus root isolation module I, magnetic lotus root isolation module II respectively, microprocessor, magnetic lotus root isolation modePiece III, serial port drive circuit connect successively, and reset circuit, jtag circuit are connected with microprocessor respectively, analogue collection moduleThe data that gather remote controller simulation rocking bar, digital data acquisition module gathers the data of locking button, Self-resetting button, and through micro-After processor is processed, through serial port drive circuit, wireless module, send to industrial computer. Described remote controller can be with 7 mouldsIntend rocking bar, i.e. engine start and stop, throttle size, a key reduction, teaching, reproduction, world coordinate system motion button, instrument controlSystem.

Described microprocessor adopts 32 TMS320F2812DSP chips, and dominant frequency can reach 150MHz.

Described magnetic lotus root isolation module is selected the ADuM1400/1/2 four-way digital isolator of ADI company. Two ends operating voltage2.7V～5.5V, supports low voltage operating and can realize level conversion.

Described robot is provided with mechanical arm, and described mechanical arm is 5DOF hydraulic machinery arm; Position sensor measurement mechanical armThe angle of each axle or length; Described mechanical arm can be by hydraulic-driven, and hydraulic power source is provided with magnetic valve and is used for starting, stopping startingThe size of machine, control throttle.

Car body move control possess advance, retreat, the function such as speed governing, left and right turn security monitoring.

Elevating mechanism motion control possesses the functions such as simple joint control, simple joint position feedback and security monitoring.

Engine and power supply power monitoring module possess the prison of the controls such as engine start and stop, speed governing and engine condition and supply voltageDepending on etc. function.

Described industrial computer adopts the small-sized industrial computer of ARK-5260, and dominant frequency 1.66GHz, with 2 network interfaces, supports 4 tunnels232/422/485 interface, carries 2 road pci expansion slots, 12-24V DC power supply.

Described receiver is exported 7 road analog signalses, voltage range 3～9V, and 6V represents meta; Export 3 way switch amount signals,Drive current 10A, 12V.

Described angle displacement sensor adopts BE1822S58,13 of overall accuracy individual pens, and the free agreement output of RS485, without looking forZero, possess starting point and demarcate and set function.

Described wireless transport module adopts SZ02-ZIGBEE wireless communication module, have that communication distance is far away, antijamming capability strong,The advantages such as networking flexibility and characteristic, can realize a point-to-multipoint and multi-multipoint between the transparent transmission of equipment room data.

Described hydraulic proportion valve adopts Danfoss valve, supply voltage 12V, and input control signal 3-9V, 6V is zero-bit.

The invention also discloses a kind of mobile robot control method, comprise the steps:

Input robot motion's initial point P₀(x₀,y₀,z₀) and terminal P_f(x_f,y_f,z_f) movement velocity P_v, Acceleration and deceleration time T_aWith interpolation cycle T_c, running time T;

The method for solving of the definite and interpolated point of basic parameter; Carrying out, before moving interpolation, determining P_vWhether meeting acceleration and deceleration wantsAsk; Method is as follows:

By P₀(x₀,y₀,z₀) and P_f(x_f,y_f,z_f) obtain actual motion distance P_d＝|P₀P_f|; By P_vAnd T_aCalculate accelerating and decelerating partRequired separation distanceIf C_d≥P_d, actual motion speedOtherwise C_v＝P_v; By time T_aWith slottingMend time T_cDraw and accelerate step number S_a; By P₀(x₀,y₀,z₀) and P_f(x_f,y_f,z_f), can obtain space the parametric equation of the straight line

$\left\{\begin{array}{c}x=k(x_f-x_0)+x_0\\ y=k(y_f-y_0)+y_0\\ z=k(z_f-z_0)+z_0\end{array}\right.$ Wherein k is scale factor, 0≤k≤1 (1)

Therefore by formula (1), can obtain each interpolated point P_i(x_i,y_i,z_i) to P₀Distance be

$C_{\mathrm{Sd}\left(i\right)}=\left|P_i{P}_0\right|=\sqrt{{(x_i-x_0)}^2+{(y_i-y_0)}^2+{(z_i-z_0)}^2}=k{P}_d$ （C_Sd(i)Represent P_i(x_i,y_i,z_i) to P₀Distance,P_d＝|P₀P_f|）(2)

I is the step number of each interpolated point, 0 withBetween all positive integers;

Making n interpolation section move distance is S_d(n), n=1 ..., i, invocation point P_iTo P₀Distance

$C_{\mathrm{Sd}\left(i\right)}=\underset{n=1}{\overset{i}{\mathrm{\Σ}}}{S}_{d\left(n\right)}=C_{\mathrm{Sd}(i-1)}+S_{d\left(i\right)},$ C_Sd(i-1)Represent P_i-1(x_i-1,y_i-1,z_i-1) to P₀Distance,

S_d(i)I interpolation section move distance, therefore it is as follows to obtain the computing formula of each interpolated point scale factor k by formula (1) and (2):

$k=\frac{C_{\mathrm{Sd}\left(i\right)}}{P_d}=\frac{\underset{n=1}{\overset{i}{\mathrm{\Σ}}}{S}_{d\left(n\right)}}{P_d}=\frac{C_{\mathrm{Sd}(i-1)}+S_{d\left(i\right)}}{P_d}$ Wherein k is scale factor, 0≤k≤1 (3)

Obtain k by formula (3), and obtain interpolated point rectangular co-ordinate.

Described accelerated motion section move distance S_d(i)The method of determining is as follows:

By actual motion speed C_vWith Acceleration and deceleration time T_aTry to achieve accelerationTherefore an acceleration Duan Shang i interpolated pointSpeed S_cv(i)＝iT_cA, obtains

$S_{d\left(i\right)}=\frac{1}{2}(S_{\mathrm{cv}\left(i\right)}+S_{\mathrm{cv}(i-1)})\·T_c=\frac{1}{2}(2i-1)a{T_c}^2---\left(4\right)$

S_cv(i-1)Represent the speed of i-1 interpolated point.

Described uniform motion section move distance S_d(i)The method of determining is as follows:

The at the uniform velocity each interpolation section move distance S of section_d(i)＝C_v·T_c。

Described retarded motion section move distance S_d(i)The method of determining is as follows:

Through after i-1 interpolated point above, obtain last distance L_d(i)＝P_d-C_Sd(i-1), therefore can obtain braking section accelerationThe speed S of a deceleration Duan Shang m interpolated point_cv(m)＝C_v+mT_cA, can obtain

$S_{d\left(m\right)}=\frac{1}{2}(S_{\mathrm{cv}\left(m\right)}+S_{\mathrm{cv}(m-1)})\·T_c=\frac{1}{2}[2C_v+a\·(2m-1)\·T_c]---\left(5\right)$

The operation principle of mobile assembly robot controller of the present invention is: gather the control of telecontrol panel by remote controller collection plateInformation processed, sends to main industrial computer with instruction type, and industrial computer gathers the information of each joint absolute value encoder simultaneously, through main workControl machine carries out, after Kinematic Algorithms processing, control instruction being sent to receiver, completes the motion control of mobile assembly robot.

Adopt such scheme, the present invention has the following advantages, and the one, the mobile assembly robot control system of designed, designed is through experimentChecking, replaces people's the arm higher automobile assembling task of frequency that fulfils assignment; The 2nd, embedded mechanical arm is little from heavy and light, size,Control system is low in energy consumption and size is little, is applicable to the application needs of mobile assembly robot. The 3rd, mobile assembly robot can be realizedComplicated a key reduction, teaching reproduction, linear interpolation motion; The 4th, system adopts modularized design, has opening, readableProperty, extensibility, maintainability, to continue exploitation. The 5th, master controller adopts main industrial computer, and main industrial computer is realized motionLearn positive and negative solution, interpolation algorithm, can realize complicated motion control, processing speed is fast. The 6th, controller band position sensorInterface, potentiometer interface, multiple functional, positional precision is high.

Brief description of the drawings

Fig. 1 is structural principle general diagram of the present invention;

Fig. 2 is remote controller collection plate theory diagram of the present invention;

Fig. 3 is vehicle-mounted mechanical arm configuration figure of the present invention;

Fig. 4 is telecontrol panel figure of the present invention;

Fig. 5 is the present invention one key reduction principle figure;

Fig. 6 is world coordinate system trajectory planning flow chart of the present invention;

Wherein, 1, remote controller; 2, wireless sending module; 3, wireless receiving module; 4, industrial computer; 5, receiver; 6,Position sensor; 7,485 buses; 8, proportioning valve; 9, switch valve; 10, analogue collection module; 11, digital data acquisitionModule; 12, magnetic lotus root isolation module I; 13, magnetic lotus root isolation module II; 14, magnetic lotus root isolation module III; 15, reset circuit; 16,Jtag circuit; 17, microprocessor; 18, serial port drive circuit; 19, crystal oscillating circuit.

Detailed description of the invention

Below in conjunction with drawings and Examples, the present invention is further described.

Referring to Fig. 1, a kind of mobile assembly robot controller, comprises remote controller 1, wireless sending module 2, wireless receiving module3, industrial computer 4, receiver 5, position sensor 6,485 buses 7, proportioning valve 8, switch valve 9.

The output of remote controller 1 control panel connects the input of data acquisition board, and the output of data acquisition board connects wireless sending module 2;The output of wireless receiving module 3 connects the input of industrial computer 4 by serial ports, the output of industrial computer 4 connects receiver 5 by serial portsInput, the output of receiver 5 connects proportioning valve 8 and switch valve 9; Position sensor 6 connects industrial computer 4 by 485 buses 7Input. The instruction of described remote controller 1, is handed down to industrial computer 4, industry control by wireless sending module 2 and wireless receiving module 3Machine 4 passes through receiver 5 downloading speed instructions, switch drive instruction to hydraulic system.

Referring to Fig. 2, described remote controller 1 collection plate comprise analogue collection module 10, digital data acquisition module 11, magnetic lotus root everyFrom module I 12, magnetic lotus root isolation module II13, magnetic lotus root isolation module III14, reset circuit 15, jtag circuit 16, microprocessor17, serial port drive circuit 18, crystal oscillating circuit 19. Analogue collection module 10 gathers the data of 7 tunnel simulation rocking bars, digital quantityAcquisition module 11 gathers the data of locking button, Self-resetting button, and after microprocessor 17 is processed, through serial port driveCircuit 18, wireless sending module 2, send to wireless receiving module 3.

Referring to Fig. 3, described mechanical arm is 5DOF, and frame for movement comprises pedestal, and support arm, support arm are installed on pedestalUpper end is shoulder joint, and forearm is arranged in shoulder joint, and the front end of forearm is connected with platform. Platform comprises platform pitch axis and platformRotating shaft.

Referring to Fig. 4, telecontrol panel schematic diagram, simulation rocking bar comprises 7, controls revolver front and back, right front and back, the large arm left side of taking turnsRight, large arm pitching, forearm are flexible, platform pitching, platform left and right; Locking button comprises instrument, selection, XYZ world coordinatesThe control of system, throttle size; Comprise the control of reset, engine start and stop, teaching, reproduction, startup from recovery button; InstructionLamp comprises power supply, radio communication, alarm lamp.

Referring to Fig. 5, mobile assembly robot is in the time of automatic motion mode, automatic according to the motion control program writing in advanceCirculation is carried out, until press the Stop button on control panel.

Referring to Fig. 6, the present embodiment adopts the mode of space line interpolation to move, and can be divided into the following steps and complete:

Input robot motion's initial point P₀(x₀,y₀,z₀) and terminal P_f(x_f,y_f,z_f) (f is the abbreviation of final) movement velocity P_v, Acceleration and deceleration time T_aWith interpolation cycle T_c, running time T;

The method for solving of the definite and interpolated point of basic parameter. Because the motion of robot space line needs through acceleration and deceleration and at the uniform velocityMotor segment, therefore carrying out, before moving interpolation, should determining P_vWhether meet acceleration and deceleration requirement. Method is as follows:

By P₀(x₀,y₀,z₀) and P_f(x_f,y_f,z_f) obtain actual motion distance P_d＝|P₀P_f|; By P_vAnd T_aCan calculate acceleration and decelerationSection required separation distanceIf C_d≥P_d, actual motion speedOtherwise C_v＝P_v; By time T_aWithInterpolation time T_cDraw and accelerate step number S_a. By P₀(x₀,y₀,z₀) and P_f(x_f,y_f,z_f), can obtain space the parametric equation of the straight line

$\left\{\begin{array}{c}x=k(x_f-x_0){+x}_0\\ y=k(y_f-y_0)+y_0\\ z=k(z_f-z_0)+z_0\end{array}\right.$ Wherein k is scale factor (0≤k≤1) (1)

Therefore by formula (1), can obtain each interpolated point P_i(x_i,y_i,z_i) (step number that i is each interpolated point, 0 withBetween all just wholeNumber) to P₀Distance be

$C_{\mathrm{Sd}\left(i\right)}=\left|P_i{P}_0\right|=\sqrt{{(x_i-x_0)}^2+{(y_i-y_0)}^2+{(z_i-z_0)}^2}=k{P}_d$ （C_Sd(i)Represent P_i(x_i,y_i,z_i) to P₀Distance,P_d＝|P₀P_f|）(2)

Making n interpolation section move distance is S_d(n)(n=1 ..., i) (n is all positive integers of 1 to i), (step that i is each interpolated pointNumber, 0 withBetween all positive integers) can invocation point P_iTo P₀Distance(C_Sd(i-1)TableShow P_i-1(x_i-1,y_i-1,z_i-1) to P₀Distance, S_d(i)I interpolation section move distance), therefore obtain each interpolated point ratio by formula (1) and (2)The computing formula of example factor k is as follows:

$k=\frac{C_{\mathrm{Sd}\left(i\right)}}{P_d}=\frac{\underset{n=1}{\overset{i}{\mathrm{\Σ}}}{S}_{d\left(n\right)}}{P_d}=\frac{C_{\mathrm{Sd}(i-1)}+S_{d\left(i\right)}}{P_d}$ Wherein k is scale factor (0≤k≤1) (3)

Just can obtain k by formula (3), and obtain interpolated point rectangular co-ordinate. Therefore space line interpolation algorithm key is to determine each interpolationSection move distance S_d(i). Introduce each section of motion below and ask for S_d(i)Method:

Accelerated motion section. Because the robot accelerating sections of the present invention's design is uniformly accelerated motion, therefore by actual motion speed C_vWith addDeceleration time T_aTry to achieve acceleration(unit is m/s^2), the therefore speed of an acceleration Duan Shang i interpolated pointS_cv(i)＝iT_cA, can obtain

$S_{d\left(i\right)}=\frac{1}{2}(S_{\mathrm{cv}\left(i\right)}+S_{\mathrm{cv}(i-1)})\·T_c=\frac{1}{2}(2i-1)a{T_c}^2$ (S_cv(i-1)Represent the speed of i-1 interpolated point) (4)

Uniform motion section. Because the robot of the present invention's design requires necessary process braking section, and interpolation operation is " calculation limit, limitWalk ", therefore carry out before uniform motion section starts, must calculating last distance and can meeting system slowdown requirement at every turn. At the uniform velocity section is each insertsMend section move distance S_d(i)＝C_v·T_c

Retarded motion section. Owing to asking for acceleration step number S_aShi Jinhang rounds calculating, therefore the simple section of will speed up accelerationAfter negate, plan accelerating sections, can introduce like this error, therefore braking section acceleration should recalculate. Through i-1 interpolated point aboveAfter, can obtain last distance L_d(i)＝P_d-C_Sd(i-1), therefore can obtain braking section accelerationDeceleration Duan Shang mThe speed S of individual interpolated point_cv(m)＝C_v+mT_cA, can obtain

$S_{d\left(m\right)}=\frac{1}{2}(S_{\mathrm{cv}\left(m\right)}+S_{\mathrm{cv}(m-1)})\·T_c=\frac{1}{2}[2C_v+a\·(2m-1)\·T_c]---\left(5\right)$

By reference to the accompanying drawings the specific embodiment of the present invention is described although above-mentioned, not the limit to protection domain of the present inventionSystem, one of ordinary skill in the art should be understood that on the basis of technical scheme of the present invention, those skilled in the art do not need to payGo out various amendments that creative work can make or distortion still in protection scope of the present invention.

Claims (5)

1. a Movement Controller of Mobile Robot, is characterized in that, comprise remote controller, radio receiving transmitting module, industrial computer, receiver,Sensor, wireless transmission system of sensor, proportioning valve, switch valve; Described remote controller is logical by radio receiving transmitting module and industrial computerLetter, industrial computer connects receiver, and receiver is connected with proportioning valve, switch valve, and sensor is located in robot, and passes through sensingDevice wireless transmitting system connects industrial computer, and the instruction of described remote controller, is handed down to industrial computer by radio receiving transmitting module, industrial computerBy the instruction of receiver downloading speed, switch drive instruction to robot; Described remote controller comprises analogue collection module, numeralAmount acquisition module, magnetic lotus root isolation module I, magnetic lotus root isolation module II, magnetic lotus root isolation module III, reset circuit, jtag circuit, micro-Processor circuit, serial port drive circuit; Analogue collection module, digital data acquisition module are respectively by magnetic lotus root isolation module I, magneticLotus root isolation module II is connected with microprocessor, and microprocessor, magnetic lotus root isolation module III, serial port drive circuit connect successively, resetsCircuit, jtag circuit are connected with microprocessor respectively, and analogue collection module gathers the data of remote controller simulation rocking bar, digital quantityAcquisition module gathers the data of locking button, Self-resetting button, and after microprocessor processes, process serial port drive circuit,Radio receiving transmitting module, sends to industrial computer.

2. a mobile robot control method with Movement Controller of Mobile Robot as claimed in claim 1, is characterized in that,Comprise the steps:

Input robot motion's initial point P₀(x₀,y₀,z₀) and terminal P_f(x_f,y_f,z_f) movement velocity P_v, Acceleration and deceleration time T_aWith interpolation cycle T_c, running time T;

The method for solving of the definite and interpolated point of basic parameter; Carrying out, before moving interpolation, determining P_vWhether meeting acceleration and deceleration wantsAsk; Method is as follows:

By P₀(x₀,y₀,z₀) and P_f(x_f,y_f,z_f) obtain actual motion distance P_d＝|P₀P_f|; By P_vAnd T_aCalculate accelerating and decelerating partRequired separation distanceIf C_d≥P_d, actual motion speedOtherwise C_v＝P_v; By time T_aWith slottingMend time T_cDraw and accelerate step number S_a; By P₀(x₀,y₀,z₀) and P_f(x_f,y_f,z_f), obtain space the parametric equation of the straight line

$\left\{\begin{array}{c}x=k(x_f-x_0)+x_0\\ y=k(y_f-y_0)+y_0\\ z=k(z_f-z_0)+z_0\end{array}\right.$ Wherein k is scale factor, 0≤k≤1 (1)

Therefore by formula (1), obtain each interpolated point P_i(x_i,y_i,z_i) to P₀Distance be

$C_{Sd\left(i\right)}=\left|P_i{P}_0\right|=\sqrt{{(x_i-x_0)}^2+{(y_i-y_0)}^2+{(z_i-z_0)}^2}={\mathrm{kP}}_d,$ C_Sd(i)Represent P_i(x_i,y_i,z_i) to P₀Distance,P_d＝|P₀P_f|，(2)

I is the step number of each interpolated point, 0 withBetween all positive integers;

Making n interpolation section move distance is S_d(n), n=1 ..., i, invocation point P_iTo P₀Distance

C_Sd(i-1)Represent P_i-1(x_i-1,y_i-1,z_i-1) to P₀Distance,

S_d(i)I interpolation section move distance, therefore it is as follows to obtain the computing formula of each interpolated point scale factor k by formula (1) and (2):

Wherein k is scale factor, 0≤k≤1, (3)

Obtain k by formula (3), and obtain interpolated point rectangular co-ordinate.

3. mobile robot control method as claimed in claim 2, is characterized in that, accelerated motion section move distance S_d(i)DetermineMethod is as follows:

By actual motion speed C_vWith Acceleration and deceleration time T_aTry to achieve accelerationTherefore an acceleration Duan Shang i interpolated pointSpeed S_cv(i)＝iT_cA, obtains

$S_{d\left(i\right)}=\frac{1}{2}(S_{cv\left(i\right)}+S_{cv(i-1)})\·T_c=\frac{1}{2}(2i-1){{\mathrm{aT}}_c}^2---\left(4\right)$

S_cv(i-1)Represent the speed of i-1 interpolated point.

4. mobile robot control method as claimed in claim 2, is characterized in that, uniform motion section move distance S_d(i)DetermineMethod is as follows:

The at the uniform velocity each interpolation section move distance S of section_d(i)＝C_v·T_c。

5. mobile robot control method as claimed in claim 2, is characterized in that, retarded motion section move distance S_d(i)DetermineMethod is as follows:

Through after i-1 interpolated point above, obtain last distance L_d(i)＝P_d-C_Sd(i-1), obtain braking section accelerationThe speed S of a deceleration Duan Shang m interpolated point_cv(m)＝C_v+mT_cA, obtains

$S_{d\left(m\right)}=\frac{1}{2}(S_{cv\left(m\right)}+S_{cv(m-1)})\·T_c=\frac{1}{2}\[2C_v+a\·(2m-1)\·T_c\]---\left(5\right).$