CN106446481B - The underactuatuated drive of pair containing flexible motion kinematics, Dynamic solving method - Google Patents
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Abstract
The present invention relates to the underactuatuated drive of pair containing flexible motion kinematics, Dynamic solving method, the step of this method, is: 1) set up mechanism coordinate system；2) mechanism parameter is defined, kinematics, the kinetic model of underactuatuated drive is established, obtains kinematics position equation, kinematics rate equation, kinematics acceleration equation and the kinetics equation of mechanism；3) each component primary condition and the driving link characteristics of motion are given, t is solved0The kinematics and kinetics equation of moment mechanism obtain the characteristics of motion of each component at this time and the driving force of driving link；4) t is solvedjThe kinematics and kinetics equation of moment mechanism obtain the characteristics of motion of each component at this time and the driving force of driving link；5) by above-mentioned steps 4) repeat n times, the driving force of the characteristics of motion and driving link of all components of the mechanism within the entire period of motion can be obtained, it is drawn point by point using MATLAB, obtains each component characteristics of motion curve graph and driving link driving force change curve.
Description
Technical field
The present invention relates to a kind of iterative numerical approach, and in particular to one kind is moved containing flexible motion pair underactuatuated drive
It learns, the method for solving of kinetics equation.
Background technique
Underactuatuated drive due to driver reduction and have many advantages, such as that low energy consumption, light weight, cost are small, but its kinematics
Imperfect with Dynamic Constraints, mechanism kinematic has uncertainty, and in the underactuatuated drive containing flexible motion pair, it is flexible
The elastic reaction of kinematic pair can play the role of driving force, make up the endless of underactuatuated drive dynamic effect to a certain extent
It is whole, but the addition of flexible motion pair also makes the kinematics of mechanism, dynamics problem become more complicated and be difficult to solve, but mechanism
Kinematics and kinetics equation Solve problems be mechanism further investigation exploitation basis, therefore the research of this kind of problem is very
It is necessary.
Underactuatuated drive containing flexible motion pair due to have simultaneously drive lacking and it is flexible of both characteristic,
Kinetic model is highly complex Nonlinear differential eguations, and it is very difficult for obtaining the analytic solutions of its dynamic response.
Currently, the kinetic model of driving compliant mechanism generally uses numerical solution entirely, with the method Step wise approximation of integral iteration, however
For the underactuatuated drive containing flexible motion pair, there are second order non-holonomic constraints for the acceleration of passive joint, have not
Integrability, therefore, traditional numerical solution are no longer applicable in.Currently, in response to this problem, there are mainly two types of solution routes: first is that from
The angle of control is set out, and is acted on using the Dynamics Coupling between main passive joint, keeps mechanism real by appropriate control strategy
It now moves, still, inward nature's rule between main passive joint is not fixed and invariable, it is with mechanism position shape and drive
The position of dynamic device and change, therefore, it is not high [1] that the shortcomings that such method, is to solve for result precision；Another method is logical
It is several subsystems with holonomic constriants that the thought of model reduction, which is crossed, by drive lacking Second Order Nonholonomic Systems depression of order, is then based on
The shortcomings that integral characteristic of each subsystem is solved, such method be need for different mechanisms be arranged different constraints into
Row depression of order, method do not have versatility [2].Therefore, for the kinematics containing flexible motion pair underactuatuated drive, power knowledge
Topic, yet there are no more perfect method of value solving.
[1]Arai H,Tachi S.Position control ofmanipulator with passive joints
using dynamic coupling[J].IEEE Transactions on Robotics&Automation,1991,7(4):
528-534.
[2]Katake A B,Mahindrakar A D,Banavar R N.Study ofunderactuated
mechanisms in the presence ofholonomic constraints:the constrained Acrobot
[C]//IEEE International Conference on Industrial Technology.2000:703-
706vol.2.
Summary of the invention
In view of the deficiencies of the prior art, the technical issues of present invention intends to solve is to provide a kind of deficient containing flexible motion pair
Driving mechanism kinematics, Dynamic solving method, this method are based on the time and speed and add when infinitely segment in two moment sections
The constant principle of speed is iterated and solves, i.e., directly thinks that velocity and acceleration is Spline smoothing in different moments section
, do not have to overcome incomplete since there are second orders in current underactuatuated drive by integration method solving speed and acceleration
Constrain and can not carry out the defect of integral iteration.
The present invention solve the technical problem the technical solution adopted is that:
The step of one kind underactuatuated drive of pair containing flexible motion kinematics, Dynamic solving method, this method, is:
1) set up mechanism coordinate system, the fixing end containing flexible motion pair underactuatuated drive are A point, are connect with fixing end
The other end of component is B point, and it is D point that flexible motion pair, which connects one end, and D point is sliding block position, the flexible motion pair other end
Be connected and fixed, using A point as origin, AD line be x-axis, be directed toward D direction be x-axis positive direction, vertical x-axis be upwards y-axis just
Direction；
2) mechanism parameter is defined, kinematics, the kinetic model of underactuatuated drive is established, does not consider each fortune in modeling process
Friction between dynamic pair, and set mechanism and be on horizontal plane, the rod piece quantity of underactuatuated drive is e, wherein driving link is denoted as i,
Remaining each rod piece is denoted as o, and the bar length of each rod piece is denoted as lm, the angle of each rod piece and x-axis is denoted as θm, wherein m=1,2 ..., e, respectively
The gravitional force of component is zero, obtain the kinematics position equation of mechanism, kinematics rate equation, kinematics acceleration equation and
Kinetics equation defines the number of iterations n, defines iteration step length δ t；
3) each component primary condition and the driving link characteristics of motion are given, t is solved0The kinematics of moment mechanism and dynamics side
Journey obtains the characteristics of motion of each component at this time and the driving force of driving link；The solution t0The kinematics of moment mechanism and dynamic
Mechanical equation carries out as follows:
The position of (3a) mechanism driving link inputs it is known that being denoted as θi(t0)；The initial position of sliding block is it is known that be denoted as x (t0),
θ is found out using kinematics position equationo(t0)；
(3b) initial time, driving link angular speedRam speedBased on step (3a), utilize
Kinematics rate equation acquires
The angular acceleration of (3c) mechanism driving link inputs it is known that being denoted asIn the basis of step (3a) and step (3b)
On the acceleration of sliding block is acquired using kinetics equation, be denoted asDriving link torque is acquired simultaneously, is denoted as τi(t0)；
(3d) will on the basis of step (3a)-(3c)WithIt brings kinematics acceleration equation into, acquires
4) it sets from tj-1Moment is to tjThe time interval at moment is infinitely small, is denoted as δ t, wherein j=1, and 2,3..., it is believed that sliding block
Velocity and acceleration it is constant in the infinitesimal section of some time interval, equal to the speed and acceleration of previous moment in section
Degree, and the overall travel speed of different moments and acceleration are Spline smoothings, solve tjThe kinematics of moment mechanism and dynamics side
Journey obtains the characteristics of motion of each component at this time and the driving force of driving link；The solution tjThe kinematics and power of moment mechanism
Equation is learned to carry out as follows:
(4a) is in tjThe position of moment mechanism driving link inputs it is known that being denoted as θi(tj)；The position of sliding block is denoted as x (tj), x
(tj) acquired by following equation,Then θ is acquired using kinematics position equationo(tj)；
The turning rate input of (4b) mechanism driving link is it is known that be denoted asThe speed of sliding block is denoted asWith tj-1Moment
It is iterated based on required result,Then it is found out using kinematics rate equation
The angular acceleration of (4c) mechanism driving link inputs it is known that being denoted asAdding for sliding block is acquired using kinetics equation
Speed and driving link torque, the acceleration of sliding block are denoted asDriving link torque is denoted as τi(tj)；
(4d) willWithKinematics acceleration equation is brought into acquire
5) by above-mentioned steps 4) repeat n times, fortune of all components of the mechanism within the entire period of motion can be obtained
The driving force of dynamic rule and driving link, is drawn point by point using MATLAB, obtains each component characteristics of motion curve graph and driving link drives
Power change curve.
Compared with prior art, the beneficial effects of the present invention are:
Present inventive concept is ingenious, proposes a kind of number containing flexible motion pair underactuatuated drive kinematics, Dynamic solving
Value method directly thinks that the velocity and acceleration of sliding block is constant in the infinitesimal section of some time interval, is equal in section
The velocity and acceleration of previous moment saves the link of Integration Solving velocity and acceleration, overcome in underactuatuated drive due to
Acceleration can not accumulate and be unable to get the defect of numerical solution, realize to the solution containing flexible motion pair underactuatuated drive.
Detailed description of the invention
Fig. 1 is the schematic diagram of mechanism containing flexible motion pair underactuatuated drive two degrees of freedom.
Fig. 2 is that slider displacement changes over time figure.
Fig. 3 is that ram speed changes over time figure.
Fig. 4 is that sliding block acceleration changes over time figure.
Fig. 5 is that driving link driving moment changes over time figure.
Fig. 6 is the schematic diagram of mechanism containing flexible motion pair underactuatuated drive Three Degree Of Freedom.
Specific embodiment
The invention will be further described with specific embodiment with reference to the accompanying drawing, embodiments of the present invention include but
It is not limited to the following example.
The present invention contains flexible motion pair underactuatuated drive (hereinafter referred to as mechanism or underactuatuated drive) kinematics, dynamics
The step of method for solving, this method, is:
1) set up mechanism coordinate system contains flexible motion pair underactuatuated drive (hereinafter referred to as mechanism or underactuatuated drive)
Fixing end is A point, and the other end for the component connecting with fixing end is B point, and it is D point that flexible motion pair, which connects one end, and D point is sliding block
Position, the flexible motion pair other end are connected and fixed, and using A point as origin, AD line is x-axis, and the direction for being directed toward D is x-axis
Positive direction, vertical x-axis are positive direction of the y-axis upwards；
2) mechanism parameter is defined, kinematics, the kinetic model of underactuatuated drive is established, does not consider each fortune in modeling process
Friction between dynamic pair, and set mechanism and be on horizontal plane, the rod piece quantity of underactuatuated drive is e, wherein driving link is denoted as i,
Remaining each rod piece is denoted as o, and the bar length of each rod piece is denoted as lm, the angle of each rod piece and x-axis is denoted as θm, wherein m=1,2 ..., e, respectively
The gravitional force of component is zero, obtain the kinematics position equation of mechanism, kinematics rate equation, kinematics acceleration equation and
Kinetics equation defines the number of iterations n, defines iteration step length δt；
3) each component primary condition and the driving link characteristics of motion are given, t is solved0The kinematics of moment mechanism and dynamics side
Journey obtains the driving force (torque) of the characteristics of motion of each component at this time and driving link；The solution t0The movement of moment mechanism
It learns and kinetics equation carries out as follows:
The position of (3a) mechanism driving link inputs it is known that being denoted as θi(t0)；The initial position of sliding block is it is known that be denoted as x (t0),
θ is found out using kinematics position equationo(t0)；
(3b) initial time, driving link angular speedRam speedBased on step (3a), utilize
Kinematics rate equation acquires
The angular acceleration of (3c) mechanism driving link inputs it is known that being denoted asIn the basis of step (3a) and step (3b)
On the acceleration of sliding block is acquired using kinetics equation, be denoted asDriving link torque is acquired simultaneously, is denoted as τi(t0)；
(3d) will on the basis of step (3a)-(3c)WithIt brings kinematics acceleration equation into, acquires
4) it sets from tj-1Moment is to tjThe time interval at moment is infinitely small, is denoted as δ t, it is believed that the velocity and acceleration of sliding block exists
It is constant in the infinitesimal section of some time interval, equal to the velocity and acceleration of previous moment in section, and different moments
Overall travel speed and acceleration are Spline smoothings, solve tjThe kinematics and kinetics equation of (j=1,2,3...) moment mechanism,
Obtain the driving force (torque) of the characteristics of motion of each component at this time and driving link；The solution tjThe kinematics of moment mechanism and dynamic
Mechanical equation carries out as follows:
(4a) is in tjThe position of (j=1,2,3...) moment mechanism driving link inputs it is known that being denoted as θi(tj)；The position of sliding block
It sets and is denoted as x (tj), x (tj) acquired by following equation,Then θ is acquired using kinematics position equationo
(tj)；
The turning rate input of (4b) mechanism driving link is it is known that be denoted asThe speed of sliding block is denoted asWith tj-1Moment
It is iterated based on required result,Then it is found out using kinematics rate equation
The angular acceleration of (4c) mechanism driving link inputs it is known that being denoted asAdding for sliding block is acquired using kinetics equation
Speed and driving link torque, the acceleration of sliding block are denoted asDriving link torque is denoted as τi(tj)；
(4d) willWithKinematics acceleration equation is brought into acquire
5) by above-mentioned steps 4) repeat n times, fortune of all components of the mechanism within the entire period of motion can be obtained
The driving force (torque) of dynamic rule and driving link, is drawn point by point using MATLAB, obtains each component characteristics of motion curve graph and original
Moving part driving force (torque) change curve.
Component in the present invention refers to rod piece and sliding block.
Further characteristic of the invention is to establish kinematics, the kinetic simulation of underactuatuated drive described in step 2)
Type, using a kind of plane two degrees of freedom underactuatuated drive containing compliant translational joint as research object, the underactuatuated drive is (referring to figure
1) there are five component, fixing end is A point, and flexible connection end is D point, one end of the sliding block connection Hookean spring at D point, linearly
The other end of spring is fixed on E point, and the tie point of first component and second component is denoted as B point, second component and third
The tie point of a component is denoted as C point, wherein and rod piece 1 is the driving link of mechanism, and rod piece 2 and rod piece 3 are remaining rod piece of mechanism,
The bar length of three rod pieces is denoted as l respectively1,l2,l3, the angle of each rod piece and x-axis is denoted as θ respectively1,θ2,θ3, the distance of A point to E point
For l0, the distance (i.e. distance of the sliding block away from origin) of A point to D point is x, and the drift of Hookean spring is a, Hookean spring it is rigid
Degree coefficient is k, and each rod piece quality is respectively m1,m2,m3, the quality of sliding block is m4,
Specific modeling procedure is as follows:
(2a) is using the kinematics position equation for closing vector method set up mechanism:
(2b) obtains the kinematics rate equation of mechanism to the kinematics position equation derivation in step (2a):
(2c) obtains the kinematics acceleration equation of mechanism to the kinematics rate equation derivation in step (2b):
(2d) obtains the kinetics equation of mechanism using Lagrange's equation method:
Wherein, q1,q2For selected generalized coordinates, J11,J12,J22It is about q1,q2Function.
Above-mentioned steps 3) described in " solve t0The kinematics and kinetics equation of moment mechanism " carries out as follows:
The position of (3a) mechanism driving link inputs it is known that being denoted as θ1(t0)；The initial position of sliding block is it is known that be denoted as x (t0)。
θ is found out using kinematics position equation2(t0),θ3(t0)；
(3b) initial time, driving link angular speedRam speedBased on step (3a), benefit
It is acquired with kinematics rate equationReferring to following formula；
The angular acceleration of (3c) mechanism driving link inputs it is known that being denoted asIn the basis of step (3a) and step (3b)
On the acceleration of sliding block is acquired using kinetics equation, be denoted asReferring to following formula
It brings the result acquired into kinetics equation simultaneously, acquires driving link torque, be denoted as τ1(t0), referring to following formula
(3d) will on the basis of step (3a)-(3c)WithKinematics acceleration equation is brought into acquire
Using following equation:
Wherein
The method that step (4a)~(4d) in step 4) solves unknown number is similar with step (3a)~(3d).
The present invention is illustrated using a kind of plane two degrees of freedom underactuatuated drive containing compliant translational joint as research object,
Obviously, the present disclosure applies equally to contain the underactuatuated drive of compliant translational joint with other of such mechanism characteristics.
Embodiment 1
The present embodiment using the plane two degrees of freedom underactuatuated drive containing compliant translational joint as research object, as shown in Figure 1,
The coordinate system of set up mechanism, taking A point is coordinate origin, and using AD line as x-axis, the direction for being directed toward D is the positive direction of x-axis, vertical x
It is positive direction of the y-axis in axial direction, each component parameter of mechanism is shown in Table 1.
The component parameter of 1 underactuatuated drive of pair containing flexible motion of table
Mechanism parameter is defined, kinematics, the kinetic model of underactuatuated drive are established, choosing iteration time step-length is
0.05s, the number of iterations are 40 times.Specific modeling procedure is as follows:
(2a) is using the kinematics position equation for closing vector method set up mechanism:
(2b) obtains the kinematics rate equation of mechanism to the kinematics position equation derivation in step (2a):
(2c) obtains the kinematics acceleration equation of mechanism to the kinematics rate equation derivation in step (2b):
(2d) obtains the kinetics equation of mechanism using Lagrange's equation method:
Wherein, q1,q2For selected generalized coordinates, J11,J12,J22It is about q1,q2Function.
For t0The kinematics of moment mechanism and the solution procedure of kinetics equation are as follows:
It is π/12, initial velocity 0rad/s, acceleration π that the characteristics of motion of (3a) given driving link, which is initial position,
rad/s2, even acceleration circular motion, the initial position of sliding block is 400mm, l0For 400mm；Bring initial position input into movement
Learn position equationθ is acquired using elimination by addition or subtraction2(t0),θ3(t0)；
(3b) initial time, driving link angular speedRam speedBy initial velocity input and step
The required amount of position known quantity and position brings equation into (3a)It utilizes
Elimination by addition or subtraction acquires
The angular acceleration of (3c) mechanism driving link inputs it is known that being denoted asBy initial acceleration input and step (3a)
With the required amount of position known quantity in step (3b) and position, speed known quantity and the required amount of speed bring kinetics equation into
It acquiresAnd bring its result into equation
Acquire driving link torque, i.e. driving torque τ1(t0)；
(3d) brings the above known quantity and acquired results into kinematics acceleration equation
It is acquired using elimination by addition or subtraction
For tjThe kinematics of moment mechanism and the solution procedure of kinetics equation are as follows:
(4a) is in tjThe position of (j=1,2,3...) moment mechanism driving link inputs it is known that being denoted as θ1(tj)；The position of sliding block
It sets and is denoted as x (tj), with tj-1It is iterated based on result required by moment, x (tj) acquired by following equation,Then θ is calculated2(tj),θ3(tj), calculation method and seek θ2(t0),θ3(t0) similar；
The turning rate input of (4b) mechanism driving link is it is known that be denoted asThe speed of sliding block is denoted asWith tj-1Moment
It is iterated based on required result,It is acquired by following equation,Then it calculatesCalculation method with askIt is similar；
The angular acceleration of (4c) mechanism driving link inputs it is known that being denoted asBy initial acceleration input and step (4a)-
The known quantity of position and speed in (4b) and required amount bring equation into
It acquiresAnd carry it into equationIt asks
Obtain driving torque τ1(tj)。
(4d) is calculatedCalculation method with askIt is similar.
(5) above-mentioned steps (4a)~(4d) 39 times is repeated, the fortune of each component in the entire period of motion can be obtained
The driving moment of dynamic rule and driving link, writes M file using matlab, obtains slide block movement rule change curve such as Fig. 2
Shown in~4, driving link driving moment change curve is as shown in Figure 5.As seen from the figure, the position of sliding block, speed, acceleration, drive
Kinetic moment curve is all more gentle, it is clear that the present invention solve the underactuatuated drive of pair containing flexible motion due to there are it is incomplete about
Beam and solve difficult problem, and 132s when entire solution procedure shares, solving speed are very fast.
Embodiment 2
The present embodiment contains compliant translational joint underactuatuated drive as research object, as shown in fig. 6, establishing machine using Three Degree Of Freedom
The coordinate system of structure, taking A point is coordinate origin, and using AD line as x-axis, the direction for being directed toward D is the positive direction of x-axis, and vertical x-axis is upward
For positive direction of the y-axis, AB rod piece and BC rod piece are driving link.Solution procedure is the same as embodiment 1.
The present invention does not address place and is suitable for the prior art.
Claims (2)
1. the step of a kind of underactuatuated drive of pair containing flexible motion kinematics, Dynamic solving method, this method, is:
1) set up mechanism coordinate system, the fixing end containing flexible motion pair underactuatuated drive are A point, the component connecting with fixing end
The other end be B point, flexible motion pair connect one end be D point, D point be sliding block position, the flexible motion pair other end connection
Fixed, using A point as origin, AD line is x-axis, and the direction for being directed toward D is the positive direction of x-axis, and vertical x-axis is positive direction of the y-axis upwards；
2) mechanism parameter is defined, kinematics, the kinetic model of underactuatuated drive is established, does not consider each kinematic pair in modeling process
Between friction, and set mechanism and be on horizontal plane, the rod piece quantity of underactuatuated drive is e, wherein driving link is denoted as i, remaining
Each rod piece is denoted as o, and the bar length of each rod piece is denoted as lm, the angle of each rod piece and x-axis is denoted as θm, wherein m=1,2 ..., e, each component
Gravitional force be zero, obtain kinematics position equation, kinematics rate equation, kinematics acceleration equation and the power of mechanism
Equation is learned, the number of iterations n is defined, defines iteration step length δ t；
3) each component primary condition and the driving link characteristics of motion are given, t is solved0The kinematics and kinetics equation of moment mechanism, obtain
To the characteristics of motion of each component at this time and the driving force of driving link；The solution t0The kinematics of moment mechanism and dynamics side
Journey carries out as follows:
The position of (3a) mechanism driving link inputs it is known that being denoted as θi(t0)；The initial position of sliding block is it is known that be denoted as x (t0), it utilizes
Kinematics position equation finds out θo(t0)；
(3b) initial time, driving link angular speedRam speedBased on step (3a), movement is utilized
Rate equation is learned to acquire
The angular acceleration of (3c) mechanism driving link inputs it is known that being denoted asIt is sharp on the basis of step (3a) and step (3b)
The acceleration that sliding block is acquired with kinetics equation, is denoted asDriving link torque is acquired simultaneously, is denoted as τi(t0)；
(3d) will on the basis of step (3a)-(3c)WithIt brings kinematics acceleration equation into, acquires
4) it sets from tj-1Moment is to tjThe time interval at moment is infinitely small, is denoted as δ t, wherein j=1, and 2,3..., it is believed that the speed of sliding block
Degree and acceleration are constant in the infinitesimal section of some time interval, equal to the velocity and acceleration of previous moment in section,
And the overall travel speed and acceleration of different moments are Spline smoothings, solve tjThe kinematics and kinetics equation of moment mechanism,
Obtain the characteristics of motion of each component at this time and the driving force of driving link；The solution tjThe kinematics of moment mechanism and dynamics side
Journey carries out as follows:
(4a) is in tjThe position of moment mechanism driving link inputs it is known that being denoted as θi(tj)；The position of sliding block is denoted as x (tj), x (tj) by
Following equation acquires,Then θ is acquired using kinematics position equationo(tj)；
The turning rate input of (4b) mechanism driving link is it is known that be denoted asThe speed of sliding block is denoted asWith tj-1Moment required knot
It is iterated based on fruit,Then it is found out using kinematics rate equation
The angular acceleration of (4c) mechanism driving link inputs it is known that being denoted asThe acceleration of sliding block is acquired using kinetics equation
And driving link torque, the acceleration of sliding block are denoted asDriving link torque is denoted as τi(tj)；
(4d) willWithKinematics acceleration equation is brought into acquire
5) by above-mentioned steps 4) repeat n times, movement rule of all components of the mechanism within the entire period of motion can be obtained
The driving force of rule and driving link, is drawn point by point using MATLAB, obtains each component characteristics of motion curve graph and driving link driving force
Change curve.
2. the underactuatuated drive of pair containing flexible motion kinematics according to claim 1, Dynamic solving method, feature exist
The kinematics for establishing underactuatuated drive described in step 2), kinetic model, with a kind of plane containing compliant translational joint
Two degrees of freedom underactuatuated drive is research object, and for the underactuatuated drive there are five component, fixing end is A point, and flexible connection end is D
Point, one end of the sliding block connection Hookean spring at D point, the other end of Hookean spring are fixed on E point, first component and second
The tie point of a component is denoted as B point, and the tie point of second component and third component is denoted as C point, wherein rod piece 1 is mechanism
Driving link, rod piece 2 and rod piece 3 are remaining rod piece of mechanism, and the bar length of three rod pieces is denoted as l respectively1,l2,l3, each rod piece and x
The angle of axis is denoted as θ respectively1,θ2,θ3, the distance of A point to E point is l0, the distance of A point to D point is x, and Hookean spring is freely grown
Degree isThe stiffness coefficient of Hookean spring is k, and each rod piece quality is respectively m1,m2,m3, the quality of sliding block is m4, specific modeling step
It is rapid as follows:
(2a) is using the kinematics position equation for closing vector method set up mechanism:
(2b) obtains the kinematics rate equation of mechanism to the kinematics position equation derivation in step (2a):
(2c) obtains the kinematics acceleration equation of mechanism to the kinematics rate equation derivation in step (2b):
(2d) obtains the kinetics equation of mechanism using Lagrange's equation method:
Wherein, q1,q2For selected generalized coordinates, J11,J12,J22It is about q1,q2Function.
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| CN103076807A (en) * | 2012-12-27 | 2013-05-01 | 北京航空航天大学 | Under-actuated flexible spacecraft attitude stabilized control method |
| CN105583832A (en) * | 2016-03-17 | 2016-05-18 | 清华大学 | Closed loop gear drive parallel-clamping and self-adapting robot finger device with flexible part |
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| CN102486225A (en) * | 2010-12-28 | 2012-06-06 | 机器时代(北京)科技有限公司 | Stay wire output structure based on under actuation |
| CN103076807A (en) * | 2012-12-27 | 2013-05-01 | 北京航空航天大学 | Under-actuated flexible spacecraft attitude stabilized control method |
| CN105583832A (en) * | 2016-03-17 | 2016-05-18 | 清华大学 | Closed loop gear drive parallel-clamping and self-adapting robot finger device with flexible part |
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| 拟人机器人手多指欠驱动机构研究;张文增;《机械设计与研究》;20071231;第23卷(第6期);第20-22、26页 |
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