Length measurement method for conical workpiece based on dual displacement sensors

A length measurement method for a conical workpiece, comprising:

• arranging a to-be-measured conical workpiece in a tooling, symmetrically arranging a first displacement sensor and a second displacement sensor at two opposite sides of the tooling, respectively;
• obtaining a distance X0 from each of the first displacement sensor and the second displacement sensor to a bottom plane of the to-be-measured conical workpiece through pre-calibration;
• arranging a base plate in an elongated shape at a tip of the to-be-measured conical workpiece;
• and
• measuring, by the first displacement sensor, a distance from the first displacement sensor to the base plate, and measuring, by the second displacement sensor, a distance from the second displacement sensor to the base plate;
• and calculating a total length X of the to-be-measured conical workpiece as according to the following formula:
• X=X0+(X1+X2)/2, wherein X1 is the distance from the first displacement sensor to the base plate, and X2 is the distance from the second displacement sensor to the base plate;
• wherein factors affecting length measurement of the to-be-measured conical workpiece comprise tooling-based calibration of a fixed length X0, measurement accuracy of the first displacement sensor and the second displacement sensor and a tilt error of the base plate;
• the tilt error of the base plate is analyzed through the following steps:
• letting the tip of the to-be-measured conical workpiece or a limited portion of the to-be-measured conical workpiece in contact with the base plate be in a sphere shape, and expressing a sphere equation as follows with a sphere center as an origin of a coordinate system:
• x2+y2+z2=R2;
• wherein R is a sphere radius, and coordinates of a vertex of the to-be-measured conical workpiece are (0, 0, R);
• considering a measurement error caused by the fact that a contact point between the to-be-measured conical workpiece and the base plate is not exactly at the vertex during actual measurement, letting an angle between the contact point and an x-axis be θ, and an angle between a projection of the contact point on a y-z plane and a y-axis be φ, such that coordinates of the contact point are expressed as:
• (x,y,z)=(R cos θ,R cos φ sin θ,R sin φ sin θ);
• wherein φ∈(0,2π), and θ is required to be as small as possible, and a tangent plane of the base plate defined by the contact point is expressed as:
• cos θx+cos φ sin θy+sin φ sin θz=R;
• letting the first displacement sensor and the second displacement sensor be both arranged on a z-axis, and coordinates of the first displacement sensor and the second displacement sensor are P1 (0,−b) and P2 (0,b), respectively, an x-axis coordinate of an intersection point between a laser emitted by the first displacement sensor and a plane of the base plate, and an x-axis coordinate of an intersection point between a laser emitted by the second displacement sensor and the plane of base plate are respectively expressed as:
• x
• 1
• =
• R
• +
• b
• ⁢
• sin
• ⁢
• φ
• ⁢
• sin
• ⁢
• θ
• cos
• ⁢
• θ
• ⁢
• and
• ⁢
• x
• 2
• =
• R
• -
• b
• ⁢
• sin
• ⁢
• φ
• ⁢
• sin
• ⁢
• θ
• cos
• ⁢
• θ
• ;
• (i) in a first case where the contact point between the plane of the base plate and the tip of the to-be-measured conical workpiece is at the vertex, θ=0, and x1=x2=R, and calculating a height h of a sphere top by averaging distances measured by the first displacement sensor and the second displacement sensor, expressed as:
• h
• =
• x
• 1
• +
• x
• 2
• 2
• =
• R
• ;
• and
• (ii) in a second case where the contact point between the plane of the base plate and the tip of the to-be-measured conical workpiece is offset from the vertex, calculating the height h of the sphere top by averaging the distances measured by the first displacement sensor and the second displacement sensor, expressed as:
• h
• =
• x
• 1
• +
• x
• 2
• 2
• =
• R
• cos
• ⁢
• θ
• ,
• wherein an error E between the height h and a true value R is expressed as:
• E
• =
• h
• -
• R
• R
• =
• 1
• -
• cos
• ⁢
• θ
• cos
• ⁢
• θ
• ;
• it is obtained that a measurement error of the first displacement sensor and the second displacement sensor is only related to the angle θ between the contact point and the x-axis, and is unrelated to the angle φ between the contact point and the y-axis;
• and
• analyzing a difference Δx between the distances measured by the first displacement sensor and the second displacement sensor, wherein a relationship between the difference Δx, the angle θ and the angle φ is expressed as:
• Δ
• ⁢
• x
• =
• x
• 1
• -
• x
• 2
• =
• 2
• ⁢
• b
• ⁢
• sin
• ⁢
• φ
• ⁢
• sin
• ⁢
• θ
• cos
• ⁢
• θ
• =
• 2
• ⁢
• b
• ⁢
• sin
• ⁢
• φ
• ⁢
• tan
• ⁢
• θ
• ;
• the difference between the distances measured by the first displacement sensor and the second displacement sensor depends on the angle θ, and an upper limit of the difference is determined by the angle θ, and the angle φ makes the difference change sinusoidally within a range of the upper limit;
• the to-be-measured conical workpiece is measured multiple times, and a minimum measured value is used as a final length measurement result.