2. Based on similar Y decomposition, a position-based visual servo control scheme is proposed. In this scheme, the end effector gradually converges to the target position, and finds the correct solution from the possible solutions during the convergence process.
In order to decompose H, the following symmetric matrix S can be considered:
It can also be written as:
So you'd better separate s:
After x and y are obtained, the analytical expressions of r, t and n are obtained after many transformations, and a summary is given in the specific expression text. At first glance, it is probably this style of painting: escape. . .
From the+/-in the expression, we can find that the solution is not unique. In fact, we have eight possible solutions. But the realistic boundary makes some solutions nonexistent. Plane π must be in front of the camera, which excludes four sets of solutions for us. It is observed that two cameras with the same point on the plane π must be on the same side of the plane (visibility constraint), which eliminates two sets of solutions.
There are many seemingly friendly numerical methods for homography decomposition, but one advantage of analytical expressions is that they can be used to derive the reciprocal formula between possible solutions, which is like this:
You can do many things with analysis and decomposition.
Position-based visual servo probably means that the robot is equipped with a visual sensor (camera), and its positioning method is to convert the pixels it sees into three-dimensional coordinates.
-First, the camera reaches a reference position and records the photos taken at that position.
-Now the servo and camera are in another place. Known information includes "photos taken by the servo at its current position" and "photos recorded at the reference position".
-The goal is not to use extra information (a? Prior) (that is, the image obtained by the camera is consistent with the servo image at the time of reference).
The input that the system can directly control is the speed of the robot. The position of the new moment can be calculated according to the position of the previous moment and the speed of the robot. ( 130)
According to the image of the current position and the image of the reference position, the matrix H can be obtained by least square estimation.
Four solutions are obtained by analytical decomposition (visibility constraints are not applied):
Using the visibility constraint, we can get two possible solutions, which may be assumed as Rtna and Rtnb (including the rotation and translation information from the reference position to the current position of the camera and the normal of the image plane).
Next, an error function (e) is defined. (adjust input according to e)
The idea is:
The weight of Rtna and Rtnb in 1. formula should be the same, because we don't know which is the real solution.
2. When the camera is in the reference position, the reference and current are in the same place, so R and T of homography should be zero, and E should be zero.
3. After finding E, define a convergence control law. When the system controls the input in this way, the error function converges to zero.
Therefore, the author puts forward the mean control law-taking the average of two possible solutions as the error function:
An intermediate step is introduced (the direction and reference position of the camera are the same, but the difference is translation error). When the error function is zero, the robot reaches the intermediate state instead of the final state.
Note that we have obtained the conversion formula between Rtna and Rtnb, so we can express the average of two possible solutions as Rtna.
The control law is defined as:
V is the input of robot, and λ can be used to adjust the convergence speed.
You can also define an update formula for the error function:
Where l is:
In this way, a closed-loop control is formed: the error function is calculated according to the current position->; (135) get the input speed->; Get a new position from speed and previous position? —& gt； (136) Calculate a new error function? —& gt； Decide on a new input.
The author also proves that under this control law, et will converge to 0 (note that it is not ta=tb=0 here, but ta+tb=0), and the module length of ta does not increase during the convergence process. Er must also converge to 0, that is, Ra=Rb. //Please fill in the corner mark otz automatically. .
Note that the above discussion assumes that the possible solutions filtered by visibility constraints are Rtna and Rtnb. However, if the possible solution is in the case of Rtna and Rtnb-, the convergence of et has not been completely proved, and the author only gives the proof that the convergence holds when the length of ta module is less than 1. (However, according to the simulation results, it seems that the condition is not satisfied (that is, when the ta modulus is greater than 1), it is also stable. )
Under the control law, when the error function is zero, R=0, and the translation error is not eliminated. In this way, we can get the correct camera orientation, and we can also get ta and tb (satisfying ta+tb = 0 and only deviating in the na direction).
In the process of obtaining this equilibrium, the na of the true solution is always the same, while the nb in the wrong solution is "on the same side" with na from the beginning (because the Rtnb solution passed the visibility constraint from the beginning). In the process of convergence, nb will gradually rotate until it becomes anti-na. This means that at some point, nb no longer satisfies the visibility constraint. Therefore, the real solution can be found in two possible solutions.
Therefore, when we detect that nb is the wrong solution, we can directly use Rtna as the E of the control input to reach the steady state more quickly. However, if we directly change from the average value of possible solutions to the correct value, E will have a mutation. It is noted that there is a partial derivative in the expression of (137) L, which will lead to errors in the control process. We need to make a gentle transition? -A switching control strategy.
When defining the error function, we don't use the average of the two directly, but use the weighting formula, namely:
Among them, the selection of weight before t-vertical detection needs to be 1, and the weight distribution changes quickly and smoothly after detection. The index seems to be a good choice. So the weight defined by the exponentially decreasing time function will be as follows:
See the effect:
Under the author's parameters, through 14 step iteration, the correct solution O.o was found.
Can I insert latex and cry? .