基于指数积公式的串联机构运动学标定方法研究
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摘要
在不过分增加制造成本的前提下提高数字化脑立体定向仪的定位精度,研究了串联机构运动学标定方法。其中,串联机构运动学参数辨识是串联机构运动学标定的核心步骤也是本文的主要研究内容。指数积公式在串联机构运动学建模方面有许多优点,但对其在运动学参数辨识方面的研究不多,主要表现为：1)基于指数积公式的误差模型中是否存在冗余参数尚未研究。2)尚未见将指数积公式引入基于运动轨迹的运动学参数辨识的研究。本文研究了指数积公式在串联机构运动学参数辨识中的实现方法和过程,进行了一系列仿真和实验。
为了明确基于指数积公式的误差模型的参数冗余性,在分析了现有指数映射关于旋量的微分求解方法之后,对一般串联机构建立了显式的误差模型。按辨识矩阵中向量之间的线性相关性,分析了误差模型中的运动学参数可辨识性,得到了3个实用的误差模型。分析结果表明,所有关节旋量坐标均可辨识；若误差模型中包含关节变量,则当所有关节旋量坐标组成的矩阵为列满秩时,关节变量可辨识。
根据指数映射和对数映射的特性,证明了在基于指数积公式的误差建模法的参数辨识算法中所采用的迭代最小二乘法收敛。
将指数积建模方法引入基于运动轨迹的运动学参数辨识,给出了实现方法,并通过实验验证了该方法的可行性和有效性。在实验中体现出优于现有同类方法,表现为可以直接从各关节运动轨迹得到运动学建模所需参数一关节旋量坐标和关节变量当量系数,而无繁琐的连杆间坐标系还原过程。
通过常见工业机器人和三坐标测量机的运动学参数辨识仿真实验,验证了3个误差模型的有效性。仿真结果表明,在理想实验条件下,它们的运动学参数均可辨识,且本文方法得到的辨识结果比其他相关方法更接近于预设真值。此外,通过仿真实验分析了测量位姿组数,测量噪声和迭代计算初值对辨识结果的影响。仿真结果表明,在测量位姿组数较少或加入测量噪声或计算初值偏离预设真值较大等不利情况下,辨识算法仍表现出良好的鲁棒性。
通过数字化脑立体定向仪标定实验,验证了基于误差建模和基于运动轨迹的串联机构运动学参数辨识方法的有效性和可操作性,给出了该方法的实现过程。经参数辨识后,数字化脑立体定向仪的绝对定位误差由1.87 mm降低到1.04 mm且距离测量误差由1.02 mm降低到0.46 mm,达到设计要求。
通过本文研究,建立了具有显式表达的指数积公式的串联机构运动学误差模型,提出了三个实用误差模型。将指数积公式建模方法引入关节运动轨迹法运动学参数辨识。阐明了关节变量零位偏差的可辨识条件。证明了基于指数积公式误差模型中最小二乘迭代算法的收敛性。仿真与实验表明,此两种串联机构运动学参数辨识方法具有通用性和实用性。
In order to improve absolute accuracy of the digitalized stereotactic apparatus without increasing manufacturing costs excessively, kinematic calibration for serial mechanisms has been investigated. As a key procedure of kinematic calibration, the kinematic parameter identification of serial mechanisms is focused on in this dissertation. Although product-of-exponential formula (POE formula) has many advantages over the other kinematic modeling methods, such few researches have given close attention to the kinematic parameter identification based on POE formula that:1) It is still not clear that whether the error model based on POE formula is involved in any redundant parameters or not.2) The POE formula has not been introduced into kinematic identification based on each joint trajectory. Not only implementation and procefure of the kinematic parameter identifcation based on POE formula was presented, but also some simulations and experiments have been conducted in this dissertation. The main contributions of this dissertation are as follows:
To clearify the identifiability of parameters in error model based on POE formula, the explicit-form error model was built for serial mechanisms after some existing approaches to solving the differential of the exponential map at twist have been analyzed. Then, the identifiability of parameters in this error model was analyzed by the linear dependence of vectors in the identification Jacobian matrix, and three practicable error models were obtained. The analysis shows that:(1) Errors in all joint twists are identifiable. (2) If the matrix formed by all joint twists is full column rank, the joint zero-position errors can be identified.
The iterative least-squares algorithm was proved to be convergent by using the characteristic of exponential map and logarithm map, which is the important part of identification algorithm based on error modeling method.
The POE formula was introduced into kinematic identification based on each joint trajectory. The procedure of kinematic identification was presented. The experiment has been conducted to demonstrate the feasibility and effectiveness of this method. This method has an advantage over the other this type of kinematic identification:joint twists and joint motion resolutions can be directly obtained from joint trajectories without the complicated procedures of the link coordinate frames'reconstruction.
Simulation results show that all kinematic parameters of common industrial robots and CMM are identifiable under the ideal experimental condition. In comparison to the simulation results in references, the identified results are closer to the true values with the suggested method. The number of measurement configurations, measurement noise and the initial values of iterative calculation, which are the influence factors on kinematic identification, are also analyzed. Although the identification algorithm was in some bad situations, such as few measurement configurations, injected measurement noise or big deviations between the computational initial values and the presetted true values, the algorithm is robust.
The experimental results of calibrating the digitalized stereotactic apparatus show that the kinematic parameter identification based on either error modeling or joint trajectory is effective and practicable. After the kinematic parameter identification, the position error and the mesuring distance error of digitalized stereotactic apparatus is reduced from 1.87 mm to 1.04 mm and from 1.02 mm to 0.46 mm respectively, which meets the requirement of design.
According to the study, three practical error models based on the exponential product formula were established, which are in explicit expressions. The exponential product formula was introduced into the kinematic parameter identification based on joint trajectories. The condition of identifying the joint zero-position errors was illustrated. The least-squares iterative algorithm using the exponential product formula based error model was proved to be convergent. Simulation and experiment results show that the two kinematic parameter identification methods are versatile and practical.
为了明确基于指数积公式的误差模型的参数冗余性,在分析了现有指数映射关于旋量的微分求解方法之后,对一般串联机构建立了显式的误差模型。按辨识矩阵中向量之间的线性相关性,分析了误差模型中的运动学参数可辨识性,得到了3个实用的误差模型。分析结果表明,所有关节旋量坐标均可辨识；若误差模型中包含关节变量,则当所有关节旋量坐标组成的矩阵为列满秩时,关节变量可辨识。
根据指数映射和对数映射的特性,证明了在基于指数积公式的误差建模法的参数辨识算法中所采用的迭代最小二乘法收敛。
将指数积建模方法引入基于运动轨迹的运动学参数辨识,给出了实现方法,并通过实验验证了该方法的可行性和有效性。在实验中体现出优于现有同类方法,表现为可以直接从各关节运动轨迹得到运动学建模所需参数一关节旋量坐标和关节变量当量系数,而无繁琐的连杆间坐标系还原过程。
通过常见工业机器人和三坐标测量机的运动学参数辨识仿真实验,验证了3个误差模型的有效性。仿真结果表明,在理想实验条件下,它们的运动学参数均可辨识,且本文方法得到的辨识结果比其他相关方法更接近于预设真值。此外,通过仿真实验分析了测量位姿组数,测量噪声和迭代计算初值对辨识结果的影响。仿真结果表明,在测量位姿组数较少或加入测量噪声或计算初值偏离预设真值较大等不利情况下,辨识算法仍表现出良好的鲁棒性。
通过数字化脑立体定向仪标定实验,验证了基于误差建模和基于运动轨迹的串联机构运动学参数辨识方法的有效性和可操作性,给出了该方法的实现过程。经参数辨识后,数字化脑立体定向仪的绝对定位误差由1.87 mm降低到1.04 mm且距离测量误差由1.02 mm降低到0.46 mm,达到设计要求。
通过本文研究,建立了具有显式表达的指数积公式的串联机构运动学误差模型,提出了三个实用误差模型。将指数积公式建模方法引入关节运动轨迹法运动学参数辨识。阐明了关节变量零位偏差的可辨识条件。证明了基于指数积公式误差模型中最小二乘迭代算法的收敛性。仿真与实验表明,此两种串联机构运动学参数辨识方法具有通用性和实用性。
In order to improve absolute accuracy of the digitalized stereotactic apparatus without increasing manufacturing costs excessively, kinematic calibration for serial mechanisms has been investigated. As a key procedure of kinematic calibration, the kinematic parameter identification of serial mechanisms is focused on in this dissertation. Although product-of-exponential formula (POE formula) has many advantages over the other kinematic modeling methods, such few researches have given close attention to the kinematic parameter identification based on POE formula that:1) It is still not clear that whether the error model based on POE formula is involved in any redundant parameters or not.2) The POE formula has not been introduced into kinematic identification based on each joint trajectory. Not only implementation and procefure of the kinematic parameter identifcation based on POE formula was presented, but also some simulations and experiments have been conducted in this dissertation. The main contributions of this dissertation are as follows:
To clearify the identifiability of parameters in error model based on POE formula, the explicit-form error model was built for serial mechanisms after some existing approaches to solving the differential of the exponential map at twist have been analyzed. Then, the identifiability of parameters in this error model was analyzed by the linear dependence of vectors in the identification Jacobian matrix, and three practicable error models were obtained. The analysis shows that:(1) Errors in all joint twists are identifiable. (2) If the matrix formed by all joint twists is full column rank, the joint zero-position errors can be identified.
The iterative least-squares algorithm was proved to be convergent by using the characteristic of exponential map and logarithm map, which is the important part of identification algorithm based on error modeling method.
The POE formula was introduced into kinematic identification based on each joint trajectory. The procedure of kinematic identification was presented. The experiment has been conducted to demonstrate the feasibility and effectiveness of this method. This method has an advantage over the other this type of kinematic identification:joint twists and joint motion resolutions can be directly obtained from joint trajectories without the complicated procedures of the link coordinate frames'reconstruction.
Simulation results show that all kinematic parameters of common industrial robots and CMM are identifiable under the ideal experimental condition. In comparison to the simulation results in references, the identified results are closer to the true values with the suggested method. The number of measurement configurations, measurement noise and the initial values of iterative calculation, which are the influence factors on kinematic identification, are also analyzed. Although the identification algorithm was in some bad situations, such as few measurement configurations, injected measurement noise or big deviations between the computational initial values and the presetted true values, the algorithm is robust.
The experimental results of calibrating the digitalized stereotactic apparatus show that the kinematic parameter identification based on either error modeling or joint trajectory is effective and practicable. After the kinematic parameter identification, the position error and the mesuring distance error of digitalized stereotactic apparatus is reduced from 1.87 mm to 1.04 mm and from 1.02 mm to 0.46 mm respectively, which meets the requirement of design.
According to the study, three practical error models based on the exponential product formula were established, which are in explicit expressions. The exponential product formula was introduced into the kinematic parameter identification based on joint trajectories. The condition of identifying the joint zero-position errors was illustrated. The least-squares iterative algorithm using the exponential product formula based error model was proved to be convergent. Simulation and experiment results show that the two kinematic parameter identification methods are versatile and practical.
引文
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