MU-2G/K
Flight Crew Commercial; Flight Crew Instrument; Flight Crew Multiengine
Flight Crew Last 90 Days 25; Flight Crew Total 2500; Flight Crew Type 600
Human-Machine Interface; Confusion; Communication Breakdown; Situational Awareness
Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued Advisory / Alert; Air Traffic Control Issued New Clearance; Flight Crew Became Reoriented
Human Factors; Procedure; Software and Automation
I was being vectored to the RNAV (GPS) Runway XXL approach to ZZZ at 4;000 feet in hazy VMC.  I received a short vector of 260 degrees to intercept the final approach course and 'join the approach.' I was flying on autopilot in heading mode and when given the instruction to 'join the approach' I pressed the approach button on my autopilot.  I had previously switched my Garmin GTN725 to VTF (Vectors To Final) mode when I began receiving vectors from ATC. Upon hitting the approach button on the AP two things happened: 1) the autopilot enabled glide path capture and 2) the autopilot made a wide turn flying through the final approach course and initiating an s-turn to track the inbound course attempting to capture the approach.  I did not hear ATC's informational statement that they would issue lower when I was clear of traffic (but I heard this when I subsequently listened to the ATC recording).While tracking back to center-line of the approach the controller informed me that I was 1 mile south of the approach course; I question that it was a full mile; so am reviewing my GPS data to ascertain exact performance of my autopilot capture versus specs for the M4D autopilot.  The autopilot specs allow an s-turn to capture; but it may not meet TERPS (Terminal Instrument Procedures) requirements if too close to the FAF and at high speed.The controller then informed me that I was below his MVA. This is apparently because I was descending per glide path but wide south of procedure course line.  I was given instructions to climb back to 4;000 feet and given a vector to start back around for another approach.  All went well from this point; and I cancelled IFR and made a visual approach to landing at ZZZ when I was realigned with the approach course.Retrospectively; there are limitations to my autopilot's capabilities and had I remained in GPSS mode and allowed my GTN725 to sequence; and resumed the approach to the next way point; turn anticipation would have made the appropriate turn onto final approach course.  Also; even though I missed that I had captured the glide path and began a descent I was not yet cleared for; staying in GPSS mode would not have allowed glide path capture.  I should not have left the assigned 4;000 feet altitude until cleared or issued another altitude.  'Join the approach' does not mean cleared for the approach.   I was always VMC; saw my traffic; and had terrain visually - none of this is an excuse; just facts.Systemic issues.  'Join the approach' is confusing wording.  I am going to go review the ATC handbook.  I am used to 'join the localizer' and know that join the localizer does not mean cleared on an ILS.  Now; thinking it through without the task pressures in the cockpit yesterday; I see that 'join the approach' does not mean cleared.  None the less; join the approach is bad wording and not clear.  I am a relatively experienced commercial instrument pilot; and this is the first time I have every been issued the instruction to 'join' an RNAV approach.  I see the parallels to joining a localizer; but a localizer is purely horizontal guidance; and we all know that.  ATC never says 'join the ILS' for example.  But saying join an RNAV approach is like saying join the ILS - it is bad wording and confusing.I learned a lot from this experience yesterday and am confident I will not repeat it.
MU-2G/K pilot reported descending below minimum vectoring altitude on approach and not realizing the autopilot was descending on glide path while the aircraft was one mile off course. Pilot was vectored back around for another approach which was conducted safely to a landing.