CN103197563A - Parallel mechanism modal space control method suitable for ophthalmic microsurgery - Google Patents

Abstract

The invention discloses a six-degrees-of-freedom parallel mechanism modal space control method suitable for parallel computing. The problem of modal decoupling matrix solving of a parallel mechanism in global motion is converted into the problem of zero values of a nonlinear system of equations; on the basis, a modal frequency and characteristic value iterative numerical algorithm is constructed based on the Newton iterative algorithm; and the iterative algorithm is high in convergence rate, few in iterative time and suitable for multi-core processor parallel computing. A global modal space controller constructed by adopting the iterative algorithm not only solves a modal transition problem in large overall motion of the six-degrees-of-freedom parallel mechanism but also improves computational efficiency of the controller greatly.

Description

Be applicable to the parallel institution mode space control method of Ophthalimic microsurgery

Technical field

The present invention relates to medical science, control and mechanical field, specifically is a kind of parallel institution mode space control method that is applicable to Ophthalimic microsurgery.

Background technology

Six-degree-of-freedom parallel connection mechanism is because it has work space not quite but the very high characteristics of control accuracy, obtained to use widely at medical domain, as calendar year 2001 medical college of the Frankfurt, Germany neurosurgery university hydrocephalus closed surgery that adopted the six-degree-of-freedom parallel robot successful implementation, the doctor is under the guiding of computer screen image, handle the action of parallel robot, this parallel robot bearing accuracy reaches 0.01mm, and what may occur when having avoided manually-operated fully trembles.After this, Germany Humboldt University medical college adopt in the laboratory Delta parallel robot also success the patient has been implemented the operation of brain section, T.Dohi etc. has developed the microoperation parallel robot that is used for cerebral surgery operation in addition, and the K.W.Grace of Northwestern university etc. has then developed the six-degree-of-freedom parallel tool hand that is used for operation on eyeball.Can be used for excision in the cataract capsule, back capsule polishing art on the six-degree-of-freedom parallel tool hand prospect, vitrectomy etc., yet for the location and track following precision of operation on eyeball and strictness thereof, parallel institution is because the Degree-of-freedom Coupling error that coupled characteristic causes has seriously limited its application in ophthalmologic operation.

Chinese scholars has proposed many control methods at the coupling influence of parallel institution, and control control method in mode space is because its explicit physical meaning, significantly expanding system frequency range and be subjected to extensive concern.This control method can be converted to multiple-input and multiple-output (MIMO) the six-degree-of-freedom parallel connection mechanism control system of strong coupling the single output of non-coupling single input (SISO) system and control, its core concept will be for will exist the degree of freedom space of strong coupling characteristic to be converted into non-coupling mode space by mode decoupling zero battle array U, thereby the traditional control theory of employing designs and controls system in the mode space.

Yet, the application of mode space controller at present is confined to parallel institution and moves among a small circle at meta, its reason is when six-degree-of-freedom parallel connection mechanism during in overall work space motion on a large scale, mode decoupling zero battle array U changes along with spatial pose, and under the influence of pose parameter, can cause the generation of mode transition phenomenon, make the change that occurs in sequence between the adjacent mode of original system.The mode transition causes after each rank of mode space controller mode control parameter and the transition the corresponding mode phenomenon that do not match, and makes mode space controller control characteristic variation even cause system oscillation.

In order to solve the mode saltus problem, a kind of overall mode space controller based on the Jacobi iteration method proposes, but this method is when handling compound attitude motion, and its speed of convergence is slower, causes controller real-time variation.

The invention provides a kind of fast convergence rate, be applicable to the parallel multiprocessor computing, the mode space controller that real-time is high.

Summary of the invention

The object of the present invention is to provide a kind of parallel institution mode space control method that is applicable to Ophthalimic microsurgery, has fast convergence rate, iterations is few, the characteristics that real-time is high, the six-degree-of-freedom parallel connection mechanism of micrurgy comprises motion platform, stationary platform, goes up to connect and cut with scissors, connect down hinge and linear actuator, adopting its core technology is a kind of real-time numerical value iterative of mode decoupling zero battle array U method that prevents the mode transition, Ω when establishing generalized frequency matrix Ω=f (sx) by the platform meta ₀=f (sx ₀) become a certain spatial pose Ω of place _T=f (sx _T), we require matrix Ω _TEigenwert decompose Ω _T=U ^T∑ U.

The present invention is achieved by the following technical solutions:

Step 1: initialization.According to six-degree-of-freedom parallel connection mechanism structural parameters r _a, r _b, α, β, h, m, I _Xx, I _Yy, I _Zzr _aBe last hinge radius of circle, r _bFor under cut with scissors radius of circle, α, β are respectively adjacent hinge minor face half central angle of lower platform, h is height of center of mass, H is podium level up and down, m is load quality, I _XxBe the moment of inertia of load around X-axis, I _YyBe the moment of inertia of load around Y-axis, I _ZzBe the moment of inertia of load around the Z axle; Calculate initial model frequency battle array ∑ and mode decoupling zero battle array U.

1) calculate initial model frequency battle array ∑:

Σ＝[λ ₁?λ ₂?λ ₃?λ ₄?λ ₅?λ ₆]

${\mathrm{\λ}}_1=\frac{6v_{1z}^2}{I_{\mathrm{zz}}}$

${\mathrm{\&lambda;}}_2=\frac{3}{2m}(\frac{m}{I_{\mathrm{xx}}}(v_{1x}^2+v_{1y}^2)+l_{n1\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+l_{n1x}^2-{({(\frac{m}{I_{\mathrm{xx}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

${\mathrm{\&lambda;}}_3=\frac{3}{2m}(\frac{m}{I_{\mathrm{xx}}}(v_{1x}^2+v_{1y}^2)+l_{n1\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+l_{n1x}^2+{({(\frac{m}{I_{\mathrm{xx}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

${\mathrm{\&lambda;}}_4=\frac{6l_{n1z}^2}{m}$

${\mathrm{\&lambda;}}_5=\frac{3}{2m}(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2+v_{1y}^2)+l_{n1\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+l_{n1x}^2-{({(\frac{m}{I_{\mathrm{yy}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{yy}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

${\mathrm{\&lambda;}}_6=\frac{3}{2m}(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2+v_{1y}^2)+l_{n1\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+l_{n1x}^2+{({(\frac{m}{I_{\mathrm{yy}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{yy}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

Wherein:

v _1x＝l _n1za _1y-l _n1ya _1z

v _1y＝l _n1xa _1z-l _n1za _1x

v _1z＝l _n1ya _1x-l _n1xa _1y

$l_{n,1}={\left[\begin{array}{ccc}{l}_{n1x}& l_{n1y}& l_{n1z}\end{array}\right]}^T={\left[\begin{array}{ccc}{r}_a\cos \mathrm{\&alpha;}-r_b\cos (\frac{\mathrm{\&pi;}}{3}-\mathrm{\&beta;})& r_a\sin \mathrm{\&alpha;}-r_b\sin (\frac{\mathrm{\&pi;}}{3}-\mathrm{\&beta;})& -H\end{array}\right]}^T/L_{\mathrm{act}}$

$L_{\mathrm{act}}=\sqrt{r_a^2+r_b^2-2r_a{r}_b\cos (\frac{\mathrm{\&pi;}}{3}-\mathrm{\&alpha;}-\mathrm{\&beta;})+{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">H</figure-callout>}}^2}$

a ₁＝a _1x?a _1y?a _1z] ^T＝[r _acosα-r _asinα?h] ^T

2) calculate initial mode decoupling zero battle array U:

U＝[u ₁?u ₂?u ₃?u ₄?u ₅?u ₆]

$\cos \mathrm{\&psi;}=\frac{t_1}{\sqrt{t_1^2+1}},\sin \mathrm{\&psi;}=\frac{1}{\sqrt{t_1^2+1}}$

$t_1=\frac{1}{2}\frac{(\frac{m}{I_{\mathrm{xx}}}(v_{1x}^2-v_{1y}^2)+l_{n1y}^2-l_{n1x}^2+{({(\frac{m}{I_{\mathrm{xx}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{1/2})}{-\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}+l_{n1x}{l}_{n1y}}$

${\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=\frac{1}{2}\frac{(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2-v_{1y}^2)+l_{n1y}^2-l_{n1x}^2+{({(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2+v_{1y}^2)+l_{n1\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+l_{n1x}^2)}^2-4{\frac{m}{I_{\mathrm{yy}}}(l_{n1x}{v}_{1x}+l_{n1y}{v}_{1y})}^2)}^{1/2})}{\frac{m}{I_{\mathrm{yy}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y}}$

Step 2: distribute each parallel processor computing initial value.

This step comprises precision ε is set, and puts iteration variable k=0, and each processor calculating initial value is set:

$x_{i,0}={\left[\begin{array}{cc}{u}_i^T& {\mathrm{\&lambda;}}_i\end{array}\right]}^T.$

Step 3: intend Newton iteration.

Each parallel processor is separate, carries out computing respectively.If distribute i processor for the treatment of i rank mode eigenwert and proper vector, adopt following formula to carry out interative computation:

x _i，k+1＝x _i，k-[F′(x _i，k)+αI] ^-1F(x _i，k)

Wherein:

$F\left(x_{i,k}\right)=\left[\begin{array}{c}{\mathrm{\&Omega;}}_T{u}_{i,k}-{\mathrm{\&lambda;}}_{i,k}{u}_{i,k}\\ u_{i,k}^T{u}_{i,k}-1\end{array}\right]$

$F^{\&prime;}\left(x_{i,k}\right)=\left[\begin{array}{cc}{\mathrm{\&Omega;}}_T-{\mathrm{\&lambda;}}_{i,k}I& -u_{i,k}\\ {\left(2u_{i,k}\right)}^{\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HSA00000878907400052.png,HSA00000878907400061.png"\; state="\{\{state\}\}">T</figure-callout>}}& 0\end{array}\right]$

α is damping factor, general desirable α=0.001.

Step 4: judge whether iteration is finished.Calculate Δ x _{I, k}=x _{I, k+1}-x _{I, k}, if || Δ x _{I, k}||≤ε, then in precision ε was set, iteration can finish explanation iteration result this moment.Otherwise get k=k+1, x _{I, k}=x _{I, k+1}Returning step 3 calculates again.

Step 5: if || Δ x _{I, k}||≤ε, take out the result: $x_{i,k+1}={\left[\begin{array}{cc}{u}_{i,k+1}^T& {\mathrm{\&lambda;}}_{i,k+1}\end{array}\right]}^T.$

Step 6: upgrade initial mode frequency lambda _iAnd modal vector u _i

u _i＝u _i，k+1，λ _i＝λ _i，k+1

Step 7: synthetic model frequency battle array ∑ and mode decoupling zero battle array U.

Σ＝[λ ₁?λ ₂?λ ₃?λ ₄?λ ₅?λ ₆]，U＝[u ₁?u ₂?u ₃?u ₄?u ₅?u ₆]

This moment, model frequency battle array ∑ and mode decoupling zero battle array U namely satisfied U ^T∑ U=Ω _T

Advantage of the present invention is:

The present invention finds the solution the null value problem that problem is converted to Nonlinear System of Equations with the modal parameter of overall work space, thought based on Newton iteration, the mode space controller of introducing the derivative information structuring has fast convergence rate, iterations is few, the characteristics that real-time is high, and be applicable to the parallel multiprocessor computing.The mode controller that the present invention proposes has been realized complete mode decoupling zero and has been eliminated because pose changes the mode transition phenomenon that causes in six-degree-of-freedom parallel connection mechanism overall situation work space, adopt the parallel institution of this mode space controller to have the control accuracy height, the characteristics that coupling error is little are applicable to the occasion that the ophthalmologic operation equal error is strict.

Description of drawings

Fig. 1 is the six-degree-of-freedom parallel connection mechanism structural representation

Fig. 2 is mode space controller block diagram

Fig. 3 is numerical value iterative algorithm process flow diagram

Each rank model frequency change curve when Fig. 4 is single dof mobility

When Fig. 5 is single dof mobility based on Jacobi iteration algorithm iteration frequency curve

Algorithm iteration frequency curve of the present invention when Fig. 6 is single dof mobility

Each rank model frequency change curve when Fig. 7 is the motion of compound degree of freedom

When Fig. 8 is the motion of compound degree of freedom based on Jacobi iteration algorithm iteration frequency curve

Algorithm iteration frequency curve of the present invention when Fig. 9 is the motion of compound degree of freedom

Embodiment

Below the invention will be further described:

Embodiment 1

Six-degree-of-freedom parallel connection mechanism comprises motion platform 1, stationary platform 2, goes up connection hinge 3, connects hinge 4 and linear actuator 5 down as shown in Figure 1.

The structure of mode space controller as shown in Figure 2, its control procedure is:

Step 1: the setting signal sx of system _DesThrough generating the preseting length signal battle array l of six linear actuators after the inverse kinematic module _Com, make difference operation with the physical length signal battle array l of six linear actuators, generate deviation matrix e, e=l _Com-l.

Step 2: deviation matrix e is carried out the mode spatial alternation, generate deviation of mode matrix e _d, e _d=U ^TE.Its mode decoupling zero battle array U obtains by overall modal calculation module, and it is relevant with the pose signal sx that the Real Time Kinematic normal solution obtains.This step is the key point of mode control, and behind the mode spatial alternation, strong coupling MIMO control system is converted into 6 nothing coupling SISO systems in the mode space, and namely the engineering staff can use familiar classic control theory that system is proofreaied and correct.

Step 3: carry out the controller design in the mode space, this moment, each rank mode controller architecture was identical with known single-degree-of-freedom control system structure, as adopting dynamic pressure FEEDBACK CONTROL etc.

Step 4: will calculate the mode driving signal matrix i of gained through mode space internal controller _dThe driving signal matrix i that is converted into actual linear actuator through the mode spatial alternation exports six drivers, i=Ui _dThe corresponding linear actuator of each driver control stretches out or the action of withdrawing, and finishes control.

As can be known, in the application of mode space controller, the most key is the calculating of mode decoupling zero battle array U, and its mathematical description is: Ω when establishing generalized frequency matrix Ω=f (sx) by the platform meta ₀=f (sx ₀) become a certain spatial pose Ω of place _T=f (sx _T), we require matrix Ω _TEigenwert decompose Ω _T=U ^T∑ U.The present invention is converted into finding the solution of nonlinear equation with this eigenwert resolution problem:

$\left\{\begin{array}{c}{\mathrm{\&Omega;}}_T{u}_i-{\mathrm{\&lambda;}}_i{u}_i=0\\ u_i^T{u}_i-1=0\end{array}\right.$ It is the problem of finding the solution of F (x)=0.

The real-time numerical value iterative of the mode decoupling zero battle array U method that the present invention proposes as shown in Figure 3.Detailed process is:

Step 1: initialization.According to six-degree-of-freedom parallel connection mechanism structural parameters r _a, r _b, α, β, h, m, I _Xx, I _Yy, I _Zzr _aBe last hinge radius of circle, r _bFor under cut with scissors radius of circle, α, β are respectively adjacent hinge minor face half central angle of lower platform, h is height of center of mass, H is podium level up and down, m is load quality, I _XxBe the moment of inertia of load around X-axis, I _YyBe the moment of inertia of load around Y-axis, I _ZzBe the moment of inertia of load around the Z axle; Calculate initial model frequency battle array ∑ and mode decoupling zero battle array U.

1) calculate initial model frequency battle array ∑:

Σ＝[λ ₁?λ ₂?λ ₃?λ ₄?λ ₅?λ ₆]

${\mathrm{\&lambda;}}_1=\frac{6v_{1z}^2}{I_{\mathrm{zz}}}$

${\mathrm{\&lambda;}}_2=\frac{3}{2m}(\frac{m}{I_{\mathrm{xx}}}(v_{1x}^2+v_{1y}^2)+l_{n1\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+l_{n1x}^2-{({(\frac{m}{I_{\mathrm{xx}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

${\mathrm{\&lambda;}}_3=\frac{3}{2m}(\frac{m}{I_{\mathrm{xx}}}(v_{1x}^2+v_{1y}^2)+l_{n1\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+l_{n1x}^2+{({(\frac{m}{I_{\mathrm{xx}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

${\mathrm{\&lambda;}}_4=\frac{6l_{n1z}^2}{m}$

${\mathrm{\&lambda;}}_5=\frac{3}{2m}(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2+v_{1y}^2)+l_{n1\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+l_{n1x}^2-{({(\frac{m}{I_{\mathrm{yy}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{yy}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

${\mathrm{\&lambda;}}_6=\frac{3}{2m}(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2+v_{1y}^2)+l_{n1\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+l_{n1x}^2+{({(\frac{m}{I_{\mathrm{yy}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{yy}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

Wherein:

v _1x＝l _n1za _1y-l _n1ya _1z

v _1y＝l _n1xa _1z-l _n1za _1x

v _1z＝l _n1ya _1x-l _n1xa _1y

$l_{n,1}={\left[\begin{array}{ccc}{l}_{n1x}& l_{n1y}& l_{n1z}\end{array}\right]}^T={\left[\begin{array}{ccc}{r}_a\cos \mathrm{\&alpha;}-r_b\cos (\frac{\mathrm{\&pi;}}{3}-\mathrm{\&beta;})& r_a\sin \mathrm{\&alpha;}-r_b\sin (\frac{\mathrm{\&pi;}}{3}-\mathrm{\&beta;})& -H\end{array}\right]}^T/L_{\mathrm{act}}$

$L_{\mathrm{act}}=\sqrt{r_a^2+r_b^2-2r_a{r}_b\cos (\frac{\mathrm{\&pi;}}{3}-\mathrm{\&alpha;}-\mathrm{\&beta;})+{\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">H</figure-callout>}}^2}$

a ₁＝[a _1x?a _1y?a _1z]T＝[r _acosα?-r _asinα?h] ^T

2) calculate initial mode decoupling zero battle array U:

U＝[u ₁?u ₂?u ₃?u ₄?u ₅?u ₆]

$\cos \mathrm{\&psi;}=\frac{t_1}{\sqrt{t_1^2+1}},\sin \mathrm{\&psi;}=\frac{1}{\sqrt{t_1^2+1}}$

$t_1=\frac{1}{2}\frac{(\frac{m}{I_{\mathrm{xx}}}(v_{1x}^2-v_{1y}^2)+l_{n1y}^2-l_{n1x}^2+{({(\frac{m}{I_{\mathrm{xx}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{1/2})}{-\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}+l_{n1x}{l}_{n1y}}$

${\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">t</figure-callout>}}_2=\frac{1}{2}\frac{(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2-v_{1y}^2)+l_{n1y}^2-l_{n1x}^2+{({(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2+v_{1y}^2)+l_{n1\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="1.png,HSA00000878907400041.png"\; state="\{\{state\}\}">y</figure-callout>}}^2+l_{n1x}^2)}^2-4{\frac{m}{I_{\mathrm{yy}}}(l_{n1x}{v}_{1x}+l_{n1y}{v}_{1y})}^2)}^{1/2})}{\frac{m}{I_{\mathrm{yy}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y}}$

Step 2: distribute each parallel processor computing initial value.

This step comprises precision ε is set, and puts iteration variable k=0, and each processor calculating initial value is set:

$x_{i,0}={\left[\begin{array}{cc}{u}_i^T& {\mathrm{\&lambda;}}_i\end{array}\right]}^T.$

Step 3: intend Newton iteration.

Each parallel processor is separate, carries out computing respectively.If distribute i processor for the treatment of i rank mode eigenwert and proper vector, adopt following formula to carry out interative computation:

x _i，k+1＝x _i，k-[F′(x _i，k)+αI] ^-1F(x _i，k)

Wherein:

$F\left(x_{i,k}\right)=\left[\begin{array}{c}{\mathrm{\&Omega;}}_T{u}_{i,k}-{\mathrm{\&lambda;}}_{i,k}{u}_{i,k}\\ u_{i,k}^T{u}_{i,k}-1\end{array}\right]$

$F^{\&prime;}\left(x_{i,k}\right)=\left[\begin{array}{cc}{\mathrm{\&Omega;}}_T-{\mathrm{\&lambda;}}_{i,k}I& -u_{i,k}\\ {\left(2u_{i,k}\right)}^{\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="HSA00000878907400052.png,HSA00000878907400061.png"\; state="\{\{state\}\}">T</figure-callout>}}& 0\end{array}\right]$

α is damping factor, and the introducing of this damping factor can solve F ' (x _{I, k}) the unusual interative computation that causes can't carry out problem, general desirable α=0.001.

Step 4: judge whether iteration is finished.Calculate Δ x _{I, k}=x _{I, k+1}-x _{I, k}, if || Δ x _{I, k}||≤ε, then in precision ε was set, iteration can finish explanation iteration result this moment.Otherwise get k=k+1, x _{I, k}=x _{I, k+1}Returning step 3 calculates again.

Step 5: if || Δ x _{I, k}||≤ε, take out the result: $x_{i,k+1}={\left[\begin{array}{cc}{u}_{i,k+1}^T& {\mathrm{\&lambda;}}_{i,k+1}\end{array}\right]}^T.$

Step 6: upgrade initial mode frequency lambda _iAnd modal vector u _i

u _i＝u _i，k+1，λ _i＝λ _i，k+1

Step 7: synthetic model frequency battle array ∑ and mode decoupling zero battle array U.

∑＝[λ ₁?λ ₂?λ ₃?λ ₄?λ ₅?λ ₆]，U＝[u ₁?u ₂?u ₃?u ₄?u ₅?u ₆]

This moment, model frequency battle array ∑ and mode decoupling zero battle array U namely satisfied U ^T∑ U=Ω _T

Embodiment:

The invention will be further described for the mode transition of employing mode space control and efficiency when carrying out compound Pose Control in conjunction with six-degree-of-freedom parallel connection mechanism in overall work space.

The platform correlation parameter is:

Podium level 2.6519m during meta, actuator travel 1.2m, on cut with scissors radius of circle 2.1148m, following hinge radius of circle 2.6519m.

The load inertial parameter is: m=13642.000kg, Ixx=46477.100kgm ², Iyy=49396.100kgm ², Izz=53865.000kgm ²

Initial mode decoupling zero battle array U:

$U=\left[\begin{array}{cccccc}0.4082& -0.4868& -0.3104& 0.4082& -0.3141& -0.4844\\ -0.4082& -0.5122& 0.2664& 0.4082& -0.2625& 0.5142\\ 0.4082& -0.0254& 0.5768& 0.4082& 0.5166& -0.0298\\ -0.4082& 0.0254& -0.5768& 0.4082& 0.5766& -0.0298\\ 0.4082& 0.5122& -0.2664& 0.4082& -0.2625& 0.5142\\ -0.4082& 0.4868& 0.3140& 0.4082& -0.3141& -0.4844\end{array}\right]$

Initial model frequency battle array ∑:

∑＝[6.7336?3.0477?9.7619?8.2085?3.0336?9.5132]Hz

Iteration precision ε: get ε=10 ^-8

For comparing, adopt simultaneously based on the mode space controller of Jacobi iteration algorithm and control, because the two has all overcome the mode transition phenomenon, so the control effect is identical, this paper compares the efficient of two kinds of control methods emphatically.

(1) single dof mobility

To making amplitude 500mm, frequency is the sinusoidal tracking of 0.1Hz to platform space attitude z:

Fig. 4 is the model frequency change curve, can see, two kinds of control algolithms have all overcome the mode transition phenomenon.

Fig. 5 is based on the required iterations of the each calculating of Jacobi iteration algorithm, can see that each calculating need can be finished for 3-4 time.

Fig. 6 can see that for algorithm of the present invention calculates required iterations at every turn each only calculating need can be finished for 2-3 time.

(2) compound degree of freedom motion

To making amplitude 100mm simultaneously, frequency is that the compounded sine of 0.1Hz is followed the tracks of to platform space attitude x and y:

Fig. 7 is the model frequency change curve.

Fig. 8 is based on the required iterations of the each calculating of Jacobi iteration algorithm, can see that each calculating need can be finished for 17-25 time.

Fig. 9 can see that for algorithm of the present invention calculates required iterations at every turn each only calculating need can be finished for 3-4 time.

Contrasting the two iterations can see, controller of the present invention its efficient under overall multiple degrees of freedom compound motion situation will be higher than the mode controller based on Jacobi iteration far away.

Claims (1)

1. parallel institution mode space control method that is applicable to Ophthalimic microsurgery, the parallel institution of Ophthalimic microsurgery comprises motion platform, stationary platform, go up and connect hinge, connect hinge and linear actuator down, generalized frequency matrix Ω=f (sx) Ω during by the platform meta ₀=f (sx ₀) become a certain spatial pose Ω of place _T=f (sx _T), require matrix Ω _TEigenwert decompose Ω _T=U ^T∑ U; It is characterized in that, adopt following steps:

Step 1: initialization, according to six-degree-of-freedom parallel connection mechanism structural parameters r _a, r _b, α, β, h, m, I _Xx, I _Yy, I _Zzr _aBe last hinge radius of circle, r _bFor under cut with scissors radius of circle, α, β are respectively adjacent hinge minor face half central angle of lower platform, h is height of center of mass, H is podium level up and down, m is load quality, I _XxBe the moment of inertia of load around X-axis, I _YyBe the moment of inertia of load around Y-axis, I _ZzBe the moment of inertia of load around the Z axle; Calculate initial model frequency battle array ∑ and mode decoupling zero battle array U;

1) calculate initial model frequency battle array ∑:

∑＝[λ ₁?λ ₂?λ ₃?λ ₄?λ ₅?λ ₆]

${\mathrm{\&lambda;}}_1=\frac{6v_{1z}^2}{I_{\mathrm{zz}}}$

${\mathrm{\&lambda;}}_2=\frac{3}{2m}(\frac{m}{I_{\mathrm{xx}}}(v_{1x}^2+v_{1y}^2)+l_{n1y}^2+l_{n1x}^2-{({(\frac{m}{I_{\mathrm{xx}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

${\mathrm{\&lambda;}}_3=\frac{3}{2m}(\frac{m}{I_{\mathrm{xx}}}(v_{1x}^2+v_{1y}^2)+l_{n1y}^2+l_{n1x}^2+{({(\frac{m}{I_{\mathrm{xx}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

${\mathrm{\&lambda;}}_4=\frac{6l_{n1z}^2}{m}$

${\mathrm{\&lambda;}}_5=\frac{3}{2m}(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2+v_{1y}^2)+l_{n1y}^2+l_{n1x}^2-{({(\frac{m}{I_{\mathrm{yy}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{yy}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

${\mathrm{\&lambda;}}_6=\frac{3}{2m}(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2+v_{1y}^2)+l_{n1y}^2+l_{n1x}^2+{({(\frac{m}{I_{\mathrm{yy}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{yy}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{\frac{1}{2}})$

Wherein:

v _1x＝l _n1za _1y-l _n1ya _1z

v _1y＝l _n1xa _1z-l _n1za _1x

v _1z＝l _n1ya _1x-l _n1xa _1y

$l_{n,1}={\left[\begin{array}{ccc}{l}_{n1x}& l_{n1y}& l_{n1z}\end{array}\right]}^T={\left[\begin{array}{ccc}{r}_a\cos \mathrm{\&alpha;}-r_b\cos (\frac{\mathrm{\&pi;}}{3}-\mathrm{\&beta;})& r_a\sin \mathrm{\&alpha;}-r_b\sin (\frac{\mathrm{\&pi;}}{3}-\mathrm{\&beta;})& -H\end{array}\right]}^T/L_{\mathrm{act}}$

$L_{\mathrm{act}}=\sqrt{r_a^2+r_b^2-2r_a{r}_b\cos (\frac{\mathrm{\&pi;}}{3}-\mathrm{\&alpha;}-\mathrm{\&beta;})+H^2}$

a ₁＝[a _1x?a _1y?a _1z] ^T＝[r _acosα?-r _asinα?h] ^T

2) calculate initial mode decoupling zero battle array U:

U＝[u ₁?u ₂?u ₃?u ₄?u ₅?u ₆]

$\cos \mathrm{\&psi;}=\frac{t_1}{\sqrt{t_1^2+1}},\sin \mathrm{\&psi;}=\frac{1}{\sqrt{t_1^2+1}}$

$t_1=\frac{1}{2}\frac{(\frac{m}{I_{\mathrm{xx}}}(v_{1x}^2-v_{1y}^2)+l_{n1y}^2-l_{n1x}^2+{({(\frac{m}{I_{\mathrm{xx}}}(v_{1y}^2-v_{1x}^2)+l_{n1x}^2-l_{n1y}^2)}^2+4{(\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y})}^2)}^{1/2})}{-\frac{m}{I_{\mathrm{xx}}}{v}_{1y}{v}_{1x}+l_{n1x}{l}_{n1y}}$

$t_2=\frac{1}{2}\frac{(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2-v_{1y}^2)+l_{n1y}^2-l_{n1x}^2+{({(\frac{m}{I_{\mathrm{yy}}}(v_{1x}^2+v_{1y}^2)+l_{n1y}^2+l_{n1x}^2)}^2-4{\frac{m}{I_{\mathrm{yy}}}(l_{n1x}{v}_{1x}+l_{n1y}{v}_{1y})}^2)}^{1/2})}{\frac{m}{I_{\mathrm{yy}}}{v}_{1y}{v}_{1x}-l_{n1x}{l}_{n1y}}$

Step 2: distribute each parallel processor computing initial value

This step comprises precision ε is set, and puts iteration variable k=0, and each processor calculating initial value is set:

$x_{i,0}={\left[\begin{array}{cc}{u}_i^T& {\mathrm{\&lambda;}}_i\end{array}\right]}^T;$

Step 3: intend Newton iteration

Each parallel processor is separate, carries out computing respectively; If distribute i processor for the treatment of i rank mode eigenwert and proper vector, adopt following formula to carry out interative computation:

x _i，k+1＝x _i，k-[F′(x _i，k)+αI] ^-1F(x _i，k)

Wherein:

$F\left(x_{i,k}\right)=\left[\begin{array}{c}{\mathrm{\&Omega;}}_T{u}_{i,k}-{\mathrm{\&lambda;}}_{i,k}{u}_{i,k}\\ u_{i,k}^T{u}_{i,k}-1\end{array}\right]$

$F^{\&prime;}\left(x_{i,k}\right)=\left[\begin{array}{cc}{\mathrm{\&Omega;}}_T-{\mathrm{\&lambda;}}_{i,k}I& -u_{i,k}\\ {\left(2u_{i,k}\right)}^T& 0\end{array}\right]$

α is damping factor, gets α=0.001;

Step 4: judge whether iteration is finished

Calculate Δ x _{I, k}=x _{I, k+1}-x _{I, k}, if || Δ x _{I, k}||≤ε, then in precision ε was set, iteration finished explanation iteration result this moment; Otherwise get k=k+1, x _{I, k}=x _{I, k+1}Returning step 3 calculates again;

Step 5: if || Δ x _{I, k}||≤ε, take out the result: $x_{i,k+1}={\left[\begin{array}{cc}{u}_{i,k+1}^T& {\mathrm{\&lambda;}}_{i,k+1}\end{array}\right]}^T;$

Step 6: upgrade initial mode frequency lambda _iAnd modal vector u _i

u _i＝u _i，k+1，λ _i＝λ _i，k+1

Step 7: synthetic model frequency battle array ∑ and mode decoupling zero battle array U;

∑＝[λ ₁?λ ₂?λ ₃?λ ₄?λ ₅?λ ₆]，U＝[u ₁?u ₂?u ₃?u ₄?u ₅?u ₆]

This moment, model frequency battle array ∑ and mode decoupling zero battle array U namely satisfied U ^T∑ U=Ω _T

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